Theorem fldextrspunlsplem 33851

Description: Lemma for fldextrspunlsp 33852: First direction. Part of the proof of Proposition 5, Chapter 5, of [BourbakiAlg2] p. 116. (Contributed by Thierry Arnoux, 13-Oct-2025.)

Hypotheses

Ref	Expression
fldextrspunfld.k	⊢ 𝐾 = (𝐿 ↾s 𝐹)
fldextrspunfld.i	⊢ 𝐼 = (𝐿 ↾s 𝐺)
fldextrspunfld.j	⊢ 𝐽 = (𝐿 ↾s 𝐻)
fldextrspunfld.2	⊢ (𝜑 → 𝐿 ∈ Field)
fldextrspunfld.3	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
fldextrspunfld.4	⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
fldextrspunfld.5	⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
fldextrspunfld.6	⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
fldextrspunlsp.n	⊢ 𝑁 = (RingSpan‘𝐿)
fldextrspunlsp.c	⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))
fldextrspunlsp.e	⊢ 𝐸 = (𝐿 ↾s 𝐶)
fldextrspunlsp.1	⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
fldextrspunlsp.2	⊢ (𝜑 → 𝐵 ∈ Fin)
fldextrspunlsplem.2	⊢ (𝜑 → 𝑃:𝐻⟶𝐺)
fldextrspunlsplem.3	⊢ (𝜑 → 𝑃 finSupp (0g‘𝐿))
fldextrspunlsplem.4	⊢ (𝜑 → 𝑋 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))))

Assertion

Ref	Expression
fldextrspunlsplem	⊢ (𝜑 → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))))

Distinct variable groups:   𝐵,𝑎,𝑏,𝑓   𝐹,𝑎,𝑏,𝑓   𝐺,𝑎,𝑓   𝐻,𝑎,𝑏,𝑓   𝐽,𝑏   𝐾,𝑎,𝑏,𝑓   𝐿,𝑎,𝑏,𝑓   𝑃,𝑎,𝑏,𝑓   𝑋,𝑎   𝜑,𝑎,𝑏,𝑓

Allowed substitution hints:   𝐶(𝑓,𝑎,𝑏)   𝐸(𝑓,𝑎,𝑏)   𝐺(𝑏)   𝐼(𝑓,𝑎,𝑏)   𝐽(𝑓,𝑎)   𝑁(𝑓,𝑎,𝑏)   𝑋(𝑓,𝑏)

Proof of Theorem fldextrspunlsplem

Dummy variables 𝑐 𝑢 𝑒 ℎ 𝑦 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fldextrspunfld.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
2	1	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐺 ∈ (SubDRing‘𝐿))
3		fldextrspunlsp.1	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
4	3	ad2antrr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
5		eqid 2737	. . . . . 6 ⊢ (0g‘𝐿) = (0g‘𝐿)
6		fldextrspunfld.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ Field)
7	6	flddrngd 20686	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ DivRing)
8	7	drngringd 20682	. . . . . . . 8 ⊢ (𝜑 → 𝐿 ∈ Ring)
9	8	ringcmnd 20231	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ CMnd)
10	9	ad3antrrr 731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐿 ∈ CMnd)
11		fldextrspunfld.6	. . . . . . 7 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
12	11	ad3antrrr 731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐻 ∈ (SubDRing‘𝐿))
13		sdrgsubrg 20736	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
14	1, 13	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
15		subrgsubg 20522	. . . . . . . 8 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ∈ (SubGrp‘𝐿))
16		subgsubm 19090	. . . . . . . 8 ⊢ (𝐺 ∈ (SubGrp‘𝐿) → 𝐺 ∈ (SubMnd‘𝐿))
17	14, 15, 16	3syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ (SubMnd‘𝐿))
18	17	ad3antrrr 731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐺 ∈ (SubMnd‘𝐿))
19		eqid 2737	. . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿)
20	14	ad3antrrr 731	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐺 ∈ (SubRing‘𝐿))
21		fldextrspunlsplem.2	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃:𝐻⟶𝐺)
22	21	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑃:𝐻⟶𝐺)
23		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ 𝐻)
24	22, 23	ffvelcdmd 7039	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑃‘𝑓) ∈ 𝐺)
25		fldextrspunfld.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
26		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐼) = (Base‘𝐼)
27	26	sdrgss 20738	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubDRing‘𝐼) → 𝐹 ⊆ (Base‘𝐼))
28	25, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐼))
29		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐿) = (Base‘𝐿)
30	29	sdrgss 20738	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
31	1, 30	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
32		fldextrspunfld.i	. . . . . . . . . . . . . 14 ⊢ 𝐼 = (𝐿 ↾s 𝐺)
33	32, 29	ressbas2 17177	. . . . . . . . . . . . 13 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘𝐼))
34	31, 33	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 = (Base‘𝐼))
35	28, 34	sseqtrrd 3973	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝐺)
36	35	ad3antrrr 731	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐹 ⊆ 𝐺)
37	3	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
38	25	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐹 ∈ (SubDRing‘𝐼))
39	11	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝐻 ∈ (SubDRing‘𝐿))
40		ovexd 7403	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝐹 ↑m 𝐵) ∈ V)
41		simpllr 776	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
42	39, 40, 41	elmaprd 32770	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
43	42, 23	ffvelcdmd 7039	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑢‘𝑓) ∈ (𝐹 ↑m 𝐵))
44	37, 38, 43	elmaprd 32770	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → (𝑢‘𝑓):𝐵⟶𝐹)
45		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → 𝑐 ∈ 𝐵)
46	44, 45	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑢‘𝑓)‘𝑐) ∈ 𝐹)
47	36, 46	sseldd 3936	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑢‘𝑓)‘𝑐) ∈ 𝐺)
48	19, 20, 24, 47	subrgmcld 33326	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) ∧ 𝑓 ∈ 𝐻) → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) ∈ 𝐺)
49	48	fmpttd 7069	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑐 ∈ 𝐵) → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))):𝐻⟶𝐺)
50	49	adantlr 716	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))):𝐻⟶𝐺)
51		fveq2 6842	. . . . . . . . 9 ⊢ (𝑓 = ℎ → (𝑃‘𝑓) = (𝑃‘ℎ))
52		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → (𝑢‘𝑓) = (𝑢‘ℎ))
53	52	fveq1d 6844	. . . . . . . . 9 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓)‘𝑐) = ((𝑢‘ℎ)‘𝑐))
54	51, 53	oveq12d 7386	. . . . . . . 8 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
55	54	cbvmptv 5204	. . . . . . 7 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
56		fvexd 6857	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (0g‘𝐿) ∈ V)
57		ssidd 3959	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐻 ⊆ 𝐻)
58		fldextrspunfld.4	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
59		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐽) = (Base‘𝐽)
60	59	sdrgss 20738	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
61	58, 60	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
62	29	sdrgss 20738	. . . . . . . . . . . . . 14 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
63	11, 62	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
64		fldextrspunfld.j	. . . . . . . . . . . . . 14 ⊢ 𝐽 = (𝐿 ↾s 𝐻)
65	64, 29	ressbas2 17177	. . . . . . . . . . . . 13 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
66	63, 65	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
67	61, 66	sseqtrrd 3973	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ⊆ 𝐻)
68	67, 63	sstrd 3946	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐿))
69	68	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐹 ⊆ (Base‘𝐿))
70	3	ad4antr 733	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
71	58	ad4antr 733	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐹 ∈ (SubDRing‘𝐽))
72		ovexd 7403	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝐹 ↑m 𝐵) ∈ V)
73		simpllr 776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
74	12, 72, 73	elmaprd 32770	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
75	74	ffvelcdmda 7038	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
76	70, 71, 75	elmaprd 32770	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ):𝐵⟶𝐹)
77		simplr 769	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝑐 ∈ 𝐵)
78	76, 77	ffvelcdmd 7039	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → ((𝑢‘ℎ)‘𝑐) ∈ 𝐹)
79	69, 78	sseldd 3936	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → ((𝑢‘ℎ)‘𝑐) ∈ (Base‘𝐿))
80	21	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑃:𝐻⟶𝐺)
81		fldextrspunlsplem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 finSupp (0g‘𝐿))
82	81	ad3antrrr 731	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑃 finSupp (0g‘𝐿))
83	8	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ 𝑦 ∈ (Base‘𝐿)) → 𝐿 ∈ Ring)
84		simpr 484	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ 𝑦 ∈ (Base‘𝐿)) → 𝑦 ∈ (Base‘𝐿))
85	29, 19, 5, 83, 84	ringlzd 20242	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ 𝑦 ∈ (Base‘𝐿)) → ((0g‘𝐿)(.r‘𝐿)𝑦) = (0g‘𝐿))
86	56, 56, 12, 57, 79, 80, 82, 85	fisuppov1 32773	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))) finSupp (0g‘𝐿))
87	55, 86	eqbrtrid 5135	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) finSupp (0g‘𝐿))
88	5, 10, 12, 18, 50, 87	gsumsubmcl 19860	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))) ∈ 𝐺)
89	88	fmpttd 7069	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))):𝐵⟶𝐺)
90	2, 4, 89	elmapdd 8790	. . 3 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) ∈ (𝐺 ↑m 𝐵))
91		breq1 5103	. . . . . 6 ⊢ (𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) → (𝑎 finSupp (0g‘𝐿) ↔ (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿)))
92	91	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → (𝑎 finSupp (0g‘𝐿) ↔ (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿)))
93		simplr 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))))
94	93	fveq1d 6844	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → (𝑎‘𝑏) = ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))‘𝑏))
95		eqid 2737	. . . . . . . . . . . 12 ⊢ (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))
96		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑏 → ((𝑢‘𝑓)‘𝑐) = ((𝑢‘𝑓)‘𝑏))
97	96	oveq2d 7384	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑏 → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)) = ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))
98	97	mpteq2dv 5194	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑏 → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))) = (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))
99	98	oveq2d 7384	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑏 → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))))
100		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
101		ovexd 7403	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑏 ∈ 𝐵) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))) ∈ V)
102	95, 99, 100, 101	fvmptd3 6973	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑏 ∈ 𝐵) → ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))‘𝑏) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))))
103	102	adantlr 716	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))‘𝑏) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))))
104	94, 103	eqtrd 2772	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → (𝑎‘𝑏) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))))
105	104	oveq1d 7383	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) ∧ 𝑏 ∈ 𝐵) → ((𝑎‘𝑏)(.r‘𝐿)𝑏) = ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))
106	105	mpteq2dva 5193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏)))
107	106	oveq2d 7384	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))))
108	107	eqeq2d 2748	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → (𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏))) ↔ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏)))))
109	92, 108	anbi12d 633	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → ((𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))) ↔ ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))))))
110	109	adantlr 716	. . 3 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑎 = (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))))) → ((𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))) ↔ ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))))))
111		fldextrspunlsp.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ Fin)
112	111	ad2antrr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐵 ∈ Fin)
113		ovexd 7403	. . . . 5 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐)))) ∈ V)
114		fvexd 6857	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (0g‘𝐿) ∈ V)
115	95, 112, 113, 114	fsuppmptdm 9291	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿))
116		fldextrspunlsplem.4	. . . . . . 7 ⊢ (𝜑 → 𝑋 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))))
117	116	ad2antrr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑋 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))))
118	8	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐿 ∈ Ring)
119	118	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐿 ∈ Ring)
120	3	ad3antrrr 731	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
121	31	ad3antrrr 731	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐺 ⊆ (Base‘𝐿))
122	21	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑃:𝐻⟶𝐺)
123	122	ffvelcdmda 7038	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑃‘ℎ) ∈ 𝐺)
124	121, 123	sseldd 3936	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑃‘ℎ) ∈ (Base‘𝐿))
125	119	adantr 480	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐿 ∈ Ring)
126	68	ad4antr 733	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐹 ⊆ (Base‘𝐿))
127	3	ad4antr 733	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
128	58	ad4antr 733	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐹 ∈ (SubDRing‘𝐽))
129	11	ad4antr 733	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝐻 ∈ (SubDRing‘𝐿))
130		ovexd 7403	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (𝐹 ↑m 𝐵) ∈ V)
131		simp-4r 784	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
132	129, 130, 131	elmaprd 32770	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
133		simplr 769	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → ℎ ∈ 𝐻)
134	132, 133	ffvelcdmd 7039	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
135	127, 128, 134	elmaprd 32770	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (𝑢‘ℎ):𝐵⟶𝐹)
136		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ 𝐵)
137	135, 136	ffvelcdmd 7039	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → ((𝑢‘ℎ)‘𝑐) ∈ 𝐹)
138	126, 137	sseldd 3936	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → ((𝑢‘ℎ)‘𝑐) ∈ (Base‘𝐿))
139		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
140		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
141	139, 140	lbsss 21041	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
142	3, 141	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
143		eqidd 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
144	143, 61	srabase 21141	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
145	66, 144	eqtr2d 2773	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘((subringAlg ‘𝐽)‘𝐹)) = 𝐻)
146	142, 145	sseqtrd 3972	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
147	146, 63	sstrd 3946	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝐿))
148	147	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐵 ⊆ (Base‘𝐿))
149	148	sselda 3935	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ (Base‘𝐿))
150	29, 19, 125, 138, 149	ringcld 20207	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐) ∈ (Base‘𝐿))
151		fvexd 6857	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (0g‘𝐿) ∈ V)
152		ssidd 3959	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐵 ⊆ 𝐵)
153	58	ad3antrrr 731	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → 𝐹 ∈ (SubDRing‘𝐽))
154	11	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐻 ∈ (SubDRing‘𝐿))
155		ovexd 7403	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐹 ↑m 𝐵) ∈ V)
156		simplr 769	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
157	154, 155, 156	elmaprd 32770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
158	157	ffvelcdmda 7038	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
159	120, 153, 158	elmaprd 32770	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ):𝐵⟶𝐹)
160	52	breq1d 5110	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓) finSupp (0g‘𝐿) ↔ (𝑢‘ℎ) finSupp (0g‘𝐿)))
161		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
162	52	fveq1d 6844	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = ℎ → ((𝑢‘𝑓)‘𝑏) = ((𝑢‘ℎ)‘𝑏))
163	162	oveq1d 7383	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = ℎ → (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))
164	163	mpteq2dv 5194	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = ℎ → (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)))
165	164	oveq2d 7384	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = ℎ → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))))
166	161, 165	eqeq12d 2753	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = ℎ → (𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))) ↔ ℎ = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)))))
167	160, 166	anbi12d 633	. . . . . . . . . . . . . 14 ⊢ (𝑓 = ℎ → (((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))) ↔ ((𝑢‘ℎ) finSupp (0g‘𝐿) ∧ ℎ = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))))))
168		simplr 769	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))))
169		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ℎ ∈ 𝐻)
170	167, 168, 169	rspcdva 3579	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ((𝑢‘ℎ) finSupp (0g‘𝐿) ∧ ℎ = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)))))
171	170	simpld 494	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑢‘ℎ) finSupp (0g‘𝐿))
172	119	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑦 ∈ (Base‘𝐿)) → 𝐿 ∈ Ring)
173		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑦 ∈ (Base‘𝐿)) → 𝑦 ∈ (Base‘𝐿))
174	29, 19, 5, 172, 173	ringlzd 20242	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑦 ∈ (Base‘𝐿)) → ((0g‘𝐿)(.r‘𝐿)𝑦) = (0g‘𝐿))
175	151, 151, 120, 152, 149, 159, 171, 174	fisuppov1 32773	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)) finSupp (0g‘𝐿))
176	29, 5, 19, 119, 120, 124, 150, 175	gsummulc2 20264	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝑃‘ℎ)(.r‘𝐿)(((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))) = ((𝑃‘ℎ)(.r‘𝐿)(𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))))
177	124	adantr 480	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (𝑃‘ℎ) ∈ (Base‘𝐿))
178	29, 19, 125, 177, 138, 149	ringassd 20204	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = ((𝑃‘ℎ)(.r‘𝐿)(((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))
179	178	mpteq2dva 5193	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐)) = (𝑐 ∈ 𝐵 ↦ ((𝑃‘ℎ)(.r‘𝐿)(((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))))
180	179	oveq2d 7384	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝑃‘ℎ)(.r‘𝐿)(((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))))
181	170	simprd 495	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ℎ = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))))
182		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → ((𝑢‘ℎ)‘𝑏) = ((𝑢‘ℎ)‘𝑐))
183		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑐 → 𝑏 = 𝑐)
184	182, 183	oveq12d 7386	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑐 → (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))
185	184	cbvmptv 5204	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏)) = (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))
186	185	oveq2i 7379	. . . . . . . . . . . 12 ⊢ (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑏)(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))
187	181, 186	eqtrdi 2788	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ℎ = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐))))
188	187	oveq2d 7384	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ((𝑃‘ℎ)(.r‘𝐿)ℎ) = ((𝑃‘ℎ)(.r‘𝐿)(𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑢‘ℎ)‘𝑐)(.r‘𝐿)𝑐)))))
189	176, 180, 188	3eqtr4rd 2783	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ℎ ∈ 𝐻) → ((𝑃‘ℎ)(.r‘𝐿)ℎ) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))
190	189	mpteq2dva 5193	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ)) = (ℎ ∈ 𝐻 ↦ (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐)))))
191	190	oveq2d 7384	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))))
192	51, 161	oveq12d 7386	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)𝑓) = ((𝑃‘ℎ)(.r‘𝐿)ℎ))
193	192	cbvmptv 5204	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓)) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ))
194	193	oveq2i 7379	. . . . . . . 8 ⊢ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ)))
195	194	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)ℎ))))
196	9	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐿 ∈ CMnd)
197	8	ad4antr 733	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐿 ∈ Ring)
198	31	ad4antr 733	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝐺 ⊆ (Base‘𝐿))
199	80	ffvelcdmda 7038	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (𝑃‘ℎ) ∈ 𝐺)
200	198, 199	sseldd 3936	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (𝑃‘ℎ) ∈ (Base‘𝐿))
201	29, 19, 197, 200, 79	ringcld 20207	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) ∈ (Base‘𝐿))
202	147	ad2antrr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝐵 ⊆ (Base‘𝐿))
203	202	sselda 3935	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝑐 ∈ (Base‘𝐿))
204	203	adantr 480	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → 𝑐 ∈ (Base‘𝐿))
205	29, 19, 197, 201, 204	ringcld 20207	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) ∈ (Base‘𝐿))
206	205	anasss 466	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ (𝑐 ∈ 𝐵 ∧ ℎ ∈ 𝐻)) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) ∈ (Base‘𝐿))
207	81	fsuppimpd 9284	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑃 supp (0g‘𝐿)) ∈ Fin)
208	207	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑃 supp (0g‘𝐿)) ∈ Fin)
209		suppssdm 8129	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 supp (0g‘𝐿)) ⊆ dom 𝑃
210	209, 21	fssdm 6689	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑃 supp (0g‘𝐿)) ⊆ 𝐻)
211	210	sseld 3934	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑓 ∈ (𝑃 supp (0g‘𝐿)) → 𝑓 ∈ 𝐻))
212	211	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → (𝑓 ∈ (𝑃 supp (0g‘𝐿)) → 𝑓 ∈ 𝐻))
213		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ (𝑢‘𝑓) finSupp (0g‘𝐿)) → (𝑢‘𝑓) finSupp (0g‘𝐿))
214	213	fsuppimpd 9284	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ (𝑢‘𝑓) finSupp (0g‘𝐿)) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin)
215	214	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → ((𝑢‘𝑓) finSupp (0g‘𝐿) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin))
216	215	adantrd 491	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → (((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin))
217	212, 216	imim12d 81	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → ((𝑓 ∈ 𝐻 → ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑓 ∈ (𝑃 supp (0g‘𝐿)) → ((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin)))
218	217	ralimdv2 3147	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) → (∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))) → ∀𝑓 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin))
219	218	imp 406	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∀𝑓 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin)
220		fveq2 6842	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑖 → (𝑢‘𝑓) = (𝑢‘𝑖))
221	220	oveq1d 7383	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑖 → ((𝑢‘𝑓) supp (0g‘𝐿)) = ((𝑢‘𝑖) supp (0g‘𝐿)))
222	221	eleq1d 2822	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑖 → (((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin ↔ ((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin))
223	222	cbvralvw 3216	. . . . . . . . . . . 12 ⊢ (∀𝑓 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑓) supp (0g‘𝐿)) ∈ Fin ↔ ∀𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin)
224	219, 223	sylib 218	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∀𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin)
225		iunfi 9255	. . . . . . . . . . 11 ⊢ (((𝑃 supp (0g‘𝐿)) ∈ Fin ∧ ∀𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin) → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin)
226	208, 224, 225	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin)
227		xpfi 9232	. . . . . . . . . 10 ⊢ ((∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) ∈ Fin ∧ (𝑃 supp (0g‘𝐿)) ∈ Fin) → (∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) × (𝑃 supp (0g‘𝐿))) ∈ Fin)
228	226, 208, 227	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) × (𝑃 supp (0g‘𝐿))) ∈ Fin)
229		snssi 4766	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ (𝑃 supp (0g‘𝐿)) → {𝑖} ⊆ (𝑃 supp (0g‘𝐿)))
230	229	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (𝑃 supp (0g‘𝐿))) → {𝑖} ⊆ (𝑃 supp (0g‘𝐿)))
231	230	iunxpssiun1 32655	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}) ⊆ (∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) × (𝑃 supp (0g‘𝐿))))
232	231	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}) ⊆ (∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))((𝑢‘𝑖) supp (0g‘𝐿)) × (𝑃 supp (0g‘𝐿))))
233	228, 232	ssfid 9181	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}) ∈ Fin)
234	21	ffnd 6671	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑃 Fn 𝐻)
235	234	ad6antr 737	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝑃 Fn 𝐻)
236	11	ad6antr 737	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐻 ∈ (SubDRing‘𝐿))
237		fvexd 6857	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (0g‘𝐿) ∈ V)
238		simpllr 776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ℎ ∈ 𝐻)
239		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ¬ ℎ ∈ (𝑃 supp (0g‘𝐿)))
240	238, 239	eldifd 3914	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ℎ ∈ (𝐻 ∖ (𝑃 supp (0g‘𝐿))))
241	235, 236, 237, 240	fvdifsupp 8123	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (𝑃‘ℎ) = (0g‘𝐿))
242	241	oveq1d 7383	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = ((0g‘𝐿)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
243	8	ad6antr 737	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐿 ∈ Ring)
244	68	ad6antr 737	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐹 ⊆ (Base‘𝐿))
245	3	ad6antr 737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
246	58	ad6antr 737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝐹 ∈ (SubDRing‘𝐽))
247		ovexd 7403	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (𝐹 ↑m 𝐵) ∈ V)
248		simp-6r 788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
249	236, 247, 248	elmaprd 32770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
250	249, 238	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
251	245, 246, 250	elmaprd 32770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → (𝑢‘ℎ):𝐵⟶𝐹)
252		simp-4r 784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → 𝑐 ∈ 𝐵)
253	251, 252	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((𝑢‘ℎ)‘𝑐) ∈ 𝐹)
254	244, 253	sseldd 3936	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((𝑢‘ℎ)‘𝑐) ∈ (Base‘𝐿))
255	29, 19, 5, 243, 254	ringlzd 20242	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((0g‘𝐿)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = (0g‘𝐿))
256	242, 255	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ ℎ ∈ (𝑃 supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = (0g‘𝐿))
257	3	ad6antr 737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
258	58	ad6antr 737	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝐹 ∈ (SubDRing‘𝐽))
259	11	ad6antr 737	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝐻 ∈ (SubDRing‘𝐿))
260		ovexd 7403	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝐹 ↑m 𝐵) ∈ V)
261		simp-6r 788	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻))
262	259, 260, 261	elmaprd 32770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝑢:𝐻⟶(𝐹 ↑m 𝐵))
263		simpllr 776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ℎ ∈ 𝐻)
264	262, 263	ffvelcdmd 7039	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝑢‘ℎ) ∈ (𝐹 ↑m 𝐵))
265	257, 258, 264	elmaprd 32770	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝑢‘ℎ):𝐵⟶𝐹)
266	265	ffnd 6671	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝑢‘ℎ) Fn 𝐵)
267		fvexd 6857	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (0g‘𝐿) ∈ V)
268		simp-4r 784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝑐 ∈ 𝐵)
269		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿)))
270	268, 269	eldifd 3914	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝑐 ∈ (𝐵 ∖ ((𝑢‘ℎ) supp (0g‘𝐿))))
271	266, 257, 267, 270	fvdifsupp 8123	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ((𝑢‘ℎ)‘𝑐) = (0g‘𝐿))
272	271	oveq2d 7384	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = ((𝑃‘ℎ)(.r‘𝐿)(0g‘𝐿)))
273	197	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → 𝐿 ∈ Ring)
274	200	ad2antrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → (𝑃‘ℎ) ∈ (Base‘𝐿))
275	29, 19, 5, 273, 274	ringrzd 20243	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)(0g‘𝐿)) = (0g‘𝐿))
276	272, 275	eqtrd 2772	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = (0g‘𝐿))
277		df-br 5101	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ ↔ ⟨𝑐, ℎ⟩ ∈ ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}))
278		fveq2 6842	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℎ = 𝑖 → (𝑢‘ℎ) = (𝑢‘𝑖))
279	278	oveq1d 7383	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑖 → ((𝑢‘ℎ) supp (0g‘𝐿)) = ((𝑢‘𝑖) supp (0g‘𝐿)))
280		sneq 4592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖})
281	279, 280	xpeq12d 5663	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℎ = 𝑖 → (((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) = (((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}))
282	281	cbviunv 4996	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ∪ ℎ ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) = ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})
283	282	eleq2i 2829	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑐, ℎ⟩ ∈ ∪ ℎ ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) ↔ ⟨𝑐, ℎ⟩ ∈ ∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖}))
284		opeliun2xp 5700	. . . . . . . . . . . . . . . . . . . 20 ⊢ (⟨𝑐, ℎ⟩ ∈ ∪ ℎ ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘ℎ) supp (0g‘𝐿)) × {ℎ}) ↔ (ℎ ∈ (𝑃 supp (0g‘𝐿)) ∧ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
285	277, 283, 284	3bitr2i 299	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ ↔ (ℎ ∈ (𝑃 supp (0g‘𝐿)) ∧ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
286	285	notbii 320	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ ↔ ¬ (ℎ ∈ (𝑃 supp (0g‘𝐿)) ∧ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
287		ianor 984	. . . . . . . . . . . . . . . . . 18 ⊢ (¬ (ℎ ∈ (𝑃 supp (0g‘𝐿)) ∧ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))) ↔ (¬ ℎ ∈ (𝑃 supp (0g‘𝐿)) ∨ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
288	286, 287	sylbb 219	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ → (¬ ℎ ∈ (𝑃 supp (0g‘𝐿)) ∨ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
289	288	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → (¬ ℎ ∈ (𝑃 supp (0g‘𝐿)) ∨ ¬ 𝑐 ∈ ((𝑢‘ℎ) supp (0g‘𝐿))))
290	256, 276, 289	mpjaodan 961	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)) = (0g‘𝐿))
291	290	oveq1d 7383	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = ((0g‘𝐿)(.r‘𝐿)𝑐))
292	118	ad3antrrr 731	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → 𝐿 ∈ Ring)
293	203	ad2antrr 727	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → 𝑐 ∈ (Base‘𝐿))
294	29, 19, 5, 292, 293	ringlzd 20242	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → ((0g‘𝐿)(.r‘𝐿)𝑐) = (0g‘𝐿))
295	291, 294	eqtrd 2772	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
296	295	an42ds 1492	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ ℎ ∈ 𝐻) ∧ 𝑐 ∈ 𝐵) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
297	296	an32s 653	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ 𝑐 ∈ 𝐵) ∧ ℎ ∈ 𝐻) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
298	297	anasss 466	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) ∧ (𝑐 ∈ 𝐵 ∧ ℎ ∈ 𝐻)) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
299	298	an32s 653	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ (𝑐 ∈ 𝐵 ∧ ℎ ∈ 𝐻)) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
300	299	anasss 466	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ ((𝑐 ∈ 𝐵 ∧ ℎ ∈ 𝐻) ∧ ¬ 𝑐∪ 𝑖 ∈ (𝑃 supp (0g‘𝐿))(((𝑢‘𝑖) supp (0g‘𝐿)) × {𝑖})ℎ)) → (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐) = (0g‘𝐿))
301	29, 5, 196, 4, 154, 206, 233, 300	gsumcom3 19919	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))))
302	191, 195, 301	3eqtr4d 2782	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))))
303	118	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → 𝐿 ∈ Ring)
304	29, 5, 19, 303, 12, 203, 201, 86	gsummulc1 20263	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) ∧ 𝑐 ∈ 𝐵) → (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))) = ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))
305	304	mpteq2dva 5193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐)))) = (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐)))
306	305	oveq2d 7384	. . . . . 6 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → (𝐿 Σg (𝑐 ∈ 𝐵 ↦ (𝐿 Σg (ℎ ∈ 𝐻 ↦ (((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))(.r‘𝐿)𝑐))))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))))
307	117, 302, 306	3eqtrd 2776	. . . . 5 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑋 = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))))
308	51, 162	oveq12d 7386	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)))
309	308	cbvmptv 5204	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)))
310	182	oveq2d 7384	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏)) = ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))
311	310	mpteq2dv 5194	. . . . . . . . . 10 ⊢ (𝑏 = 𝑐 → (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))
312	309, 311	eqtrid 2784	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))) = (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))
313	312	oveq2d 7384	. . . . . . . 8 ⊢ (𝑏 = 𝑐 → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏)))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐)))))
314	313, 183	oveq12d 7386	. . . . . . 7 ⊢ (𝑏 = 𝑐 → ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏) = ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))
315	314	cbvmptv 5204	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏)) = (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐))
316	315	oveq2i 7379	. . . . 5 ⊢ (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑐 ∈ 𝐵 ↦ ((𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑃‘ℎ)(.r‘𝐿)((𝑢‘ℎ)‘𝑐))))(.r‘𝐿)𝑐)))
317	307, 316	eqtr4di 2790	. . . 4 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏))))
318	115, 317	jca 511	. . 3 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ((𝑐 ∈ 𝐵 ↦ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑐))))) finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑃‘𝑓)(.r‘𝐿)((𝑢‘𝑓)‘𝑏))))(.r‘𝐿)𝑏)))))
319	90, 110, 318	rspcedvd 3580	. 2 ⊢ (((𝜑 ∧ 𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)) ∧ ∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))) → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))))
320		breq1 5103	. . . 4 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑒 finSupp (0g‘𝐿) ↔ (𝑢‘𝑓) finSupp (0g‘𝐿)))
321		fveq1 6841	. . . . . . . 8 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑒‘𝑏) = ((𝑢‘𝑓)‘𝑏))
322	321	oveq1d 7383	. . . . . . 7 ⊢ (𝑒 = (𝑢‘𝑓) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) = (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))
323	322	mpteq2dv 5194	. . . . . 6 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)) = (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))
324	323	oveq2d 7384	. . . . 5 ⊢ (𝑒 = (𝑢‘𝑓) → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))
325	324	eqeq2d 2748	. . . 4 ⊢ (𝑒 = (𝑢‘𝑓) → (𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))) ↔ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))))
326	320, 325	anbi12d 633	. . 3 ⊢ (𝑒 = (𝑢‘𝑓) → ((𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)))) ↔ ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏))))))
327		ovexd 7403	. . 3 ⊢ (𝜑 → (𝐹 ↑m 𝐵) ∈ V)
328		eqid 2737	. . . . . . . . . 10 ⊢ (LSpan‘((subringAlg ‘𝐽)‘𝐹)) = (LSpan‘((subringAlg ‘𝐽)‘𝐹))
329	139, 140, 328	lbssp 21043	. . . . . . . . 9 ⊢ (𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
330	3, 329	syl 17	. . . . . . . 8 ⊢ (𝜑 → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
331	144, 66, 330	3eqtr4rd 2783	. . . . . . 7 ⊢ (𝜑 → ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) = 𝐻)
332	331	eleq2d 2823	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) ↔ 𝑓 ∈ 𝐻))
333		eqid 2737	. . . . . . 7 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹)))
334		eqid 2737	. . . . . . 7 ⊢ (Scalar‘((subringAlg ‘𝐽)‘𝐹)) = (Scalar‘((subringAlg ‘𝐽)‘𝐹))
335		eqid 2737	. . . . . . 7 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹)))
336		eqid 2737	. . . . . . 7 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))
337		sdrgsubrg 20736	. . . . . . . . 9 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ∈ (SubRing‘𝐽))
338	58, 337	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubRing‘𝐽))
339		eqid 2737	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹)
340	339	sralmod 21151	. . . . . . . 8 ⊢ (𝐹 ∈ (SubRing‘𝐽) → ((subringAlg ‘𝐽)‘𝐹) ∈ LMod)
341	338, 340	syl 17	. . . . . . 7 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) ∈ LMod)
342	328, 139, 333, 334, 335, 336, 341, 142	ellspds 33461	. . . . . 6 ⊢ (𝜑 → (𝑓 ∈ ((LSpan‘((subringAlg ‘𝐽)‘𝐹))‘𝐵) ↔ ∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))))
343	332, 342	bitr3d 281	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ 𝐻 ↔ ∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))))
344	343	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐻) → ∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)))))
345		eqid 2737	. . . . . . . . . 10 ⊢ (𝐽 ↾s 𝐹) = (𝐽 ↾s 𝐹)
346	345, 59	ressbas2 17177	. . . . . . . . 9 ⊢ (𝐹 ⊆ (Base‘𝐽) → 𝐹 = (Base‘(𝐽 ↾s 𝐹)))
347	61, 346	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 = (Base‘(𝐽 ↾s 𝐹)))
348	143, 61	srasca 21144	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾s 𝐹) = (Scalar‘((subringAlg ‘𝐽)‘𝐹)))
349	348	fveq2d 6846	. . . . . . . 8 ⊢ (𝜑 → (Base‘(𝐽 ↾s 𝐹)) = (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))))
350	347, 349	eqtr2d 2773	. . . . . . 7 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = 𝐹)
351	350	oveq1d 7383	. . . . . 6 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵) = (𝐹 ↑m 𝐵))
352		sdrgsubrg 20736	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ∈ (SubRing‘𝐿))
353	11, 352	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ (SubRing‘𝐿))
354		subrgsubg 20522	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubRing‘𝐿) → 𝐻 ∈ (SubGrp‘𝐿))
355	64, 5	subg0 19074	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubGrp‘𝐿) → (0g‘𝐿) = (0g‘𝐽))
356	353, 354, 355	3syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐿) = (0g‘𝐽))
357	64	sdrgdrng 20735	. . . . . . . . . . . . . . 15 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐽 ∈ DivRing)
358	11, 357	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐽 ∈ DivRing)
359	358	drngringd 20682	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ Ring)
360	359	ringcmnd 20231	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ CMnd)
361	360	cmnmndd 19745	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ Mnd)
362		subrgsubg 20522	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubRing‘𝐽) → 𝐹 ∈ (SubGrp‘𝐽))
363		eqid 2737	. . . . . . . . . . . . 13 ⊢ (0g‘𝐽) = (0g‘𝐽)
364	363	subg0cl 19076	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (SubGrp‘𝐽) → (0g‘𝐽) ∈ 𝐹)
365	338, 362, 364	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (0g‘𝐽) ∈ 𝐹)
366	345, 59, 363	ress0g 18699	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Mnd ∧ (0g‘𝐽) ∈ 𝐹 ∧ 𝐹 ⊆ (Base‘𝐽)) → (0g‘𝐽) = (0g‘(𝐽 ↾s 𝐹)))
367	361, 365, 61, 366	syl3anc 1374	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘𝐽) = (0g‘(𝐽 ↾s 𝐹)))
368	348	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝜑 → (0g‘(𝐽 ↾s 𝐹)) = (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))))
369	356, 367, 368	3eqtrrd 2777	. . . . . . . . 9 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) = (0g‘𝐿))
370	369	breq2d 5112	. . . . . . . 8 ⊢ (𝜑 → (𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↔ 𝑒 finSupp (0g‘𝐿)))
371	370	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↔ 𝑒 finSupp (0g‘𝐿)))
372	3	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
373		subgsubm 19090	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ (SubGrp‘𝐿) → 𝐻 ∈ (SubMnd‘𝐿))
374	353, 354, 373	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐻 ∈ (SubMnd‘𝐿))
375	374	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐻 ∈ (SubMnd‘𝐿))
376	64, 19	ressmulr 17239	. . . . . . . . . . . . . . . 16 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → (.r‘𝐿) = (.r‘𝐽))
377	11, 376	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (.r‘𝐿) = (.r‘𝐽))
378	143, 61	sravsca 21145	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (.r‘𝐽) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
379	377, 378	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
380	379	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)))
381	380	oveqd 7385	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) = ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))
382	353	ad2antrr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝐻 ∈ (SubRing‘𝐿))
383	67	ad2antrr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝐹 ⊆ 𝐻)
384	25	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐹 ∈ (SubDRing‘𝐼))
385	351	eleq2d 2823	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵) ↔ 𝑒 ∈ (𝐹 ↑m 𝐵)))
386	385	biimpa 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝑒 ∈ (𝐹 ↑m 𝐵))
387	372, 384, 386	elmaprd 32770	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝑒:𝐵⟶𝐹)
388	387	ffvelcdmda 7038	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝑒‘𝑏) ∈ 𝐹)
389	383, 388	sseldd 3936	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝑒‘𝑏) ∈ 𝐻)
390	146	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → 𝐵 ⊆ 𝐻)
391	390	sselda 3935	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐻)
392	19, 382, 389, 391	subrgmcld 33326	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)(.r‘𝐿)𝑏) ∈ 𝐻)
393	381, 392	eqeltrrd 2838	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏) ∈ 𝐻)
394	393	fmpttd 7069	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)):𝐵⟶𝐻)
395	372, 375, 394, 64	gsumsubm 18772	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (𝐽 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))
396	377, 378	eqtr2d 2773	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = (.r‘𝐿))
397	396	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → ( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹)) = (.r‘𝐿))
398	397	oveqd 7385	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏) = ((𝑒‘𝑏)(.r‘𝐿)𝑏))
399	398	mpteq2dv 5194	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)) = (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)))
400	399	oveq2d 7384	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))
401	3	mptexd 7180	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)) ∈ V)
402		fvexd 6857	. . . . . . . . . . 11 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) ∈ V)
403	339, 401, 358, 402, 61	gsumsra 33141	. . . . . . . . . 10 ⊢ (𝜑 → (𝐽 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))
404	403	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝐽 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))))
405	395, 400, 404	3eqtr3rd 2781	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))
406	405	eqeq2d 2748	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → (𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏))) ↔ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)))))
407	371, 406	anbi12d 633	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)) → ((𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)))) ↔ (𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))))
408	351, 407	rexeqbidva 3305	. . . . 5 ⊢ (𝜑 → (∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)))) ↔ ∃𝑒 ∈ (𝐹 ↑m 𝐵)(𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))))
409	408	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐻) → (∃𝑒 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ↑m 𝐵)(𝑒 finSupp (0g‘(Scalar‘((subringAlg ‘𝐽)‘𝐹))) ∧ 𝑓 = (((subringAlg ‘𝐽)‘𝐹) Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)( ·𝑠 ‘((subringAlg ‘𝐽)‘𝐹))𝑏)))) ↔ ∃𝑒 ∈ (𝐹 ↑m 𝐵)(𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏))))))
410	344, 409	mpbid 232	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐻) → ∃𝑒 ∈ (𝐹 ↑m 𝐵)(𝑒 finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑒‘𝑏)(.r‘𝐿)𝑏)))))
411	326, 11, 327, 410	ac6mapd 32713	. 2 ⊢ (𝜑 → ∃𝑢 ∈ ((𝐹 ↑m 𝐵) ↑m 𝐻)∀𝑓 ∈ 𝐻 ((𝑢‘𝑓) finSupp (0g‘𝐿) ∧ 𝑓 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ (((𝑢‘𝑓)‘𝑏)(.r‘𝐿)𝑏)))))
412	319, 411	r19.29a 3146	1 ⊢ (𝜑 → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑋 = (𝐿 Σg (𝑏 ∈ 𝐵 ↦ ((𝑎‘𝑏)(.r‘𝐿)𝑏)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ∪ cun 3901   ⊆ wss 3903  {csn 4582  ⟨cop 4588  ∪ ciun 4948   class class class wbr 5100   ↦ cmpt 5181   × cxp 5630   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   supp csupp 8112   ↑_m cmap 8775  Fincfn 8895   finSupp cfsupp 9276  Basecbs 17148   ↾_s cress 17169  ._rcmulr 17190  Scalarcsca 17192   ·_𝑠 cvsca 17193  0_gc0g 17371   Σ_g cgsu 17372  Mndcmnd 18671  SubMndcsubmnd 18719  SubGrpcsubg 19062  CMndccmn 19721  Ringcrg 20180  SubRingcsubrg 20514  RingSpancrgspn 20555  DivRingcdr 20674  Fieldcfield 20675  SubDRingcsdrg 20731  LModclmod 20823  LSpanclspn 20934  LBasisclbs 21038  subringAlg csra 21135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-reg 9509  ax-inf2 9562  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632  df-om 7819  df-1st 7943  df-2nd 7944  df-supp 8113  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-ixp 8848  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9277  df-sup 9357  df-oi 9427  df-r1 9688  df-rank 9689  df-card 9863  df-ac 10038  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-fzo 13583  df-seq 13937  df-hash 14266  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-mulr 17203  df-sca 17205  df-vsca 17206  df-ip 17207  df-tset 17208  df-ple 17209  df-ds 17211  df-hom 17213  df-cco 17214  df-0g 17373  df-gsum 17374  df-prds 17379  df-pws 17381  df-mre 17517  df-mrc 17518  df-acs 17520  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-mhm 18720  df-submnd 18721  df-grp 18878  df-minusg 18879  df-sbg 18880  df-mulg 19010  df-subg 19065  df-ghm 19154  df-cntz 19258  df-cmn 19723  df-abl 19724  df-mgp 20088  df-rng 20100  df-ur 20129  df-ring 20182  df-nzr 20458  df-subrng 20491  df-subrg 20515  df-drng 20676  df-field 20677  df-sdrg 20732  df-lmod 20825  df-lss 20895  df-lsp 20935  df-lmhm 20986  df-lbs 21039  df-sra 21137  df-rgmod 21138  df-dsmm 21699  df-frlm 21714  df-uvc 21750

This theorem is referenced by:  fldextrspunlsp  33852