Theorem fldextrspunlsp 34005

Description: Lemma for fldextrspunfld 34007. The subring generated by the union of two field extensions 𝐺 and 𝐻 is the vector sub- 𝐺 space generated by a basis 𝐵 of 𝐻. 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)

Assertion

Ref	Expression
fldextrspunlsp	⊢ (𝜑 → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵))

Proof of Theorem fldextrspunlsp

Dummy variables 𝑎 𝑓 𝑔 ℎ 𝑝 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fldextrspunlsp.c	. . . . 5 ⊢ 𝐶 = (𝑁‘(𝐺 ∪ 𝐻))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐶 = (𝑁‘(𝐺 ∪ 𝐻)))
3	2	eleq2d 2855	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ (𝑁‘(𝐺 ∪ 𝐻))))
4		eqid 2769	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
5		eqid 2769	. . . 4 ⊢ (.r‘𝐿) = (.r‘𝐿)
6		eqid 2769	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
7		fldextrspunlsp.n	. . . 4 ⊢ 𝑁 = (RingSpan‘𝐿)
8		fldextrspunfld.2	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Field)
9	8	fldcrngd 20822	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
10		fldextrspunfld.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
11		sdrgsubrg 20868	. . . . 5 ⊢ (𝐺 ∈ (SubDRing‘𝐿) → 𝐺 ∈ (SubRing‘𝐿))
12	10, 11	syl 18	. . . 4 ⊢ (𝜑 → 𝐺 ∈ (SubRing‘𝐿))
13		fldextrspunfld.6	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (SubDRing‘𝐿))
14		sdrgsubrg 20868	. . . . 5 ⊢ (𝐻 ∈ (SubDRing‘𝐿) → 𝐻 ∈ (SubRing‘𝐿))
15	13, 14	syl 18	. . . 4 ⊢ (𝜑 → 𝐻 ∈ (SubRing‘𝐿))
16	4, 5, 6, 7, 9, 12, 15	elrgspnsubrun 33506	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑁‘(𝐺 ∪ 𝐻)) ↔ ∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))))
17	4	subrgss 20653	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
18	12, 17	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
19		eqid 2769	. . . . . . . . 9 ⊢ (𝐿 ↾s 𝐺) = (𝐿 ↾s 𝐺)
20	19, 4	ressbas2 17294	. . . . . . . 8 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘(𝐿 ↾s 𝐺)))
21	18, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (Base‘(𝐿 ↾s 𝐺)))
22		eqidd 2770	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺))
23	22, 18	srasca 21275	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ↾s 𝐺) = (Scalar‘((subringAlg ‘𝐿)‘𝐺)))
24	23	fveq2d 6883	. . . . . . 7 ⊢ (𝜑 → (Base‘(𝐿 ↾s 𝐺)) = (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))))
25	21, 24	eqtr2d 2805	. . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = 𝐺)
26	25	oveq1d 7423	. . . . 5 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↑m 𝐵) = (𝐺 ↑m 𝐵))
27	9	crngringd 20324	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ Ring)
28	27	ringcmnd 20363	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ CMnd)
29	28	cmnmndd 19870	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Mnd)
30		subrgsubg 20658	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ∈ (SubGrp‘𝐿))
31	12, 30	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝐿))
32	6	subg0cl 19196	. . . . . . . . . 10 ⊢ (𝐺 ∈ (SubGrp‘𝐿) → (0g‘𝐿) ∈ 𝐺)
33	31, 32	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐿) ∈ 𝐺)
34	19, 4, 6	ress0g 18816	. . . . . . . . 9 ⊢ ((𝐿 ∈ Mnd ∧ (0g‘𝐿) ∈ 𝐺 ∧ 𝐺 ⊆ (Base‘𝐿)) → (0g‘𝐿) = (0g‘(𝐿 ↾s 𝐺)))
35	29, 33, 18, 34	syl3anc 1396	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐿) = (0g‘(𝐿 ↾s 𝐺)))
36	23	fveq2d 6883	. . . . . . . 8 ⊢ (𝜑 → (0g‘(𝐿 ↾s 𝐺)) = (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))))
37	35, 36	eqtr2d 2805	. . . . . . 7 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (0g‘𝐿))
38	37	breq2d 5122	. . . . . 6 ⊢ (𝜑 → (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↔ 𝑎 finSupp (0g‘𝐿)))
39		eqid 2769	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺)
40		fldextrspunlsp.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
41	40	mptexd 7220	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)) ∈ V)
42	39	sralmod 21282	. . . . . . . . . 10 ⊢ (𝐺 ∈ (SubRing‘𝐿) → ((subringAlg ‘𝐿)‘𝐺) ∈ LMod)
43	12, 42	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐺) ∈ LMod)
44	39, 41, 8, 43, 18	gsumsra 33304	. . . . . . . 8 ⊢ (𝜑 → (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))) = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
45	22, 18	sravsca 21276	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺)))
46	45	oveqd 7425	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑎‘𝑣)(.r‘𝐿)𝑣) = ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))
47	46	mpteq2dv 5206	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)) = (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣)))
48	47	oveq2d 7424	. . . . . . . 8 ⊢ (𝜑 → (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))) = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))))
49	44, 48	eqtr2d 2805	. . . . . . 7 ⊢ (𝜑 → (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))) = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
50	49	eqeq2d 2780	. . . . . 6 ⊢ (𝜑 → (𝑥 = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))) ↔ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))))
51	38, 50	anbi12d 643	. . . . 5 ⊢ (𝜑 → ((𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ∧ 𝑥 = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣)))) ↔ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))))
52	26, 51	rexeqbidv 3346	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↑m 𝐵)(𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ∧ 𝑥 = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣)))) ↔ ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))))
53		eqid 2769	. . . . 5 ⊢ (LSpan‘((subringAlg ‘𝐿)‘𝐺)) = (LSpan‘((subringAlg ‘𝐿)‘𝐺))
54		eqid 2769	. . . . 5 ⊢ (Base‘((subringAlg ‘𝐿)‘𝐺)) = (Base‘((subringAlg ‘𝐿)‘𝐺))
55		eqid 2769	. . . . 5 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺)))
56		eqid 2769	. . . . 5 ⊢ (Scalar‘((subringAlg ‘𝐿)‘𝐺)) = (Scalar‘((subringAlg ‘𝐿)‘𝐺))
57		eqid 2769	. . . . 5 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺)))
58		eqid 2769	. . . . 5 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺)) = ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))
59		eqid 2769	. . . . . . . . . 10 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
60		eqid 2769	. . . . . . . . . 10 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
61	59, 60	lbsss 21172	. . . . . . . . 9 ⊢ (𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
62	40, 61	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((subringAlg ‘𝐽)‘𝐹)))
63	4	subrgss 20653	. . . . . . . . . . 11 ⊢ (𝐻 ∈ (SubRing‘𝐿) → 𝐻 ⊆ (Base‘𝐿))
64	15, 63	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐻 ⊆ (Base‘𝐿))
65		fldextrspunfld.j	. . . . . . . . . . 11 ⊢ 𝐽 = (𝐿 ↾s 𝐻)
66	65, 4	ressbas2 17294	. . . . . . . . . 10 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
67	64, 66	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
68		eqidd 2770	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
69		fldextrspunfld.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
70		eqid 2769	. . . . . . . . . . . 12 ⊢ (Base‘𝐽) = (Base‘𝐽)
71	70	sdrgss 20870	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
72	69, 71	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
73	68, 72	srabase 21272	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
74	67, 73	eqtrd 2804	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
75	62, 74	sseqtrrd 3982	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
76	75, 64	sstrd 3955	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝐿))
77	22, 18	srabase 21272	. . . . . 6 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐺)))
78	76, 77	sseqtrd 3981	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((subringAlg ‘𝐿)‘𝐺)))
79	53, 54, 55, 56, 57, 58, 43, 78	ellspds 33622	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↑m 𝐵)(𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ∧ 𝑥 = (((subringAlg ‘𝐿)‘𝐺) Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))))))
80		fldextrspunfld.k	. . . . . . 7 ⊢ 𝐾 = (𝐿 ↾s 𝐹)
81		fldextrspunfld.i	. . . . . . 7 ⊢ 𝐼 = (𝐿 ↾s 𝐺)
82	8	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐿 ∈ Field)
83		fldextrspunfld.3	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐼))
84	83	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐹 ∈ (SubDRing‘𝐼))
85	69	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐹 ∈ (SubDRing‘𝐽))
86	10	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐺 ∈ (SubDRing‘𝐿))
87	13	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐻 ∈ (SubDRing‘𝐿))
88		fldextrspunlsp.e	. . . . . . 7 ⊢ 𝐸 = (𝐿 ↾s 𝐶)
89	40	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
90		fldextrspunlsp.2	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin)
91	90	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐵 ∈ Fin)
92		simplr 780	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑝 ∈ (𝐺 ↑m 𝐻))
93	87, 86, 92	elmaprd 32962	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑝:𝐻⟶𝐺)
94		simprl 782	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑝 finSupp (0g‘𝐿))
95		simprr 784	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))
96		fveq2 6879	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑝‘𝑓) = (𝑝‘ℎ))
97		id 23	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
98	96, 97	oveq12d 7426	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑝‘𝑓)(.r‘𝐿)𝑓) = ((𝑝‘ℎ)(.r‘𝐿)ℎ))
99	98	cbvmptv 5216	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)) = (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ))
100	99	oveq2i 7419	. . . . . . . 8 ⊢ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ)))
101	95, 100	eqtrdi 2820	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝑥 = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ))))
102	80, 81, 65, 82, 84, 85, 86, 87, 7, 1, 88, 89, 91, 93, 94, 101	fldextrspunlsplem 34004	. . . . . 6 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))))
103	102	r19.29an 3175	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))))
104		breq1 5113	. . . . . . . 8 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝑝 finSupp (0g‘𝐿) ↔ (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) finSupp (0g‘𝐿)))
105		fveq1 6878	. . . . . . . . . . . 12 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝑝‘𝑓) = ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓))
106	105	oveq1d 7423	. . . . . . . . . . 11 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → ((𝑝‘𝑓)(.r‘𝐿)𝑓) = (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))
107	106	mpteq2dv 5206	. . . . . . . . . 10 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)) = (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)))
108	107	oveq2d 7424	. . . . . . . . 9 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))
109	108	eqeq2d 2780	. . . . . . . 8 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → (𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))) ↔ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)))))
110	104, 109	anbi12d 643	. . . . . . 7 ⊢ (𝑝 = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) → ((𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))) ↔ ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))))
111	10	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝐺 ∈ (SubDRing‘𝐿))
112	13	ad2antrr 738	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝐻 ∈ (SubDRing‘𝐿))
113	40	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
114	10	adantr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝐺 ∈ (SubDRing‘𝐿))
115		simpr 489	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝑎 ∈ (𝐺 ↑m 𝐵))
116	113, 114, 115	elmaprd 32962	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝑎:𝐵⟶𝐺)
117	116	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ 𝐻) → 𝑎:𝐵⟶𝐺)
118	117	ffvelcdmda 7077	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ 𝐺)
119	33	ad4antr 744	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ 𝐻) ∧ ¬ 𝑔 ∈ 𝐵) → (0g‘𝐿) ∈ 𝐺)
120	118, 119	ifclda 4525	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ 𝐻) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) ∈ 𝐺)
121	120	fmpttd 7108	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))):𝐻⟶𝐺)
122	111, 112, 121	elmapdd 8834	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) ∈ (𝐺 ↑m 𝐻))
123		fvexd 6894	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → (0g‘𝐿) ∈ V)
124	121	ffund 6708	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → Fun (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))))
125		simprl 782	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝑎 finSupp (0g‘𝐿))
126	116	ffnd 6704	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝑎 Fn 𝐵)
127	126	ad3antrrr 742	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝑎 Fn 𝐵)
128	40	ad4antr 744	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
129		fvexd 6894	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → (0g‘𝐿) ∈ V)
130		simpr 489	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵)
131		simplr 780	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿))))
132	131	eldifbd 3926	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → ¬ 𝑔 ∈ (𝑎 supp (0g‘𝐿)))
133	130, 132	eldifd 3924	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿))))
134	127, 128, 129, 133	fvdifsupp 8163	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) = (0g‘𝐿))
135		eqidd 2770	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ ¬ 𝑔 ∈ 𝐵) → (0g‘𝐿) = (0g‘𝐿))
136	134, 135	ifeqda 4526	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) ∧ 𝑔 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) = (0g‘𝐿))
137	136, 112	suppss2 8192	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) supp (0g‘𝐿)) ⊆ (𝑎 supp (0g‘𝐿)))
138	122, 123, 124, 125, 137	fsuppsssuppgd 9338	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) finSupp (0g‘𝐿))
139		eqid 2769	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))
140		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑔 = 𝑓)
141		suppssdm 8169	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 supp (0g‘𝐿)) ⊆ dom 𝑎
142	116	fdmd 6714	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → dom 𝑎 = 𝐵)
143	142	adantr 485	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → dom 𝑎 = 𝐵)
144	141, 143	sseqtrid 3987	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ⊆ 𝐵)
145	144	sselda 3945	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → 𝑓 ∈ 𝐵)
146	145	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑓 ∈ 𝐵)
147	140, 146	eqeltrd 2869	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑔 ∈ 𝐵)
148	147	iftrued 4497	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) = (𝑎‘𝑔))
149		fveq2 6879	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑓 → (𝑎‘𝑔) = (𝑎‘𝑓))
150	149	adantl 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → (𝑎‘𝑔) = (𝑎‘𝑓))
151	148, 150	eqtrd 2804	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) = (𝑎‘𝑓))
152	75	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐵 ⊆ 𝐻)
153	144, 152	sstrd 3955	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ⊆ 𝐻)
154	153	sselda 3945	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → 𝑓 ∈ 𝐻)
155		fvexd 6894	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → (𝑎‘𝑓) ∈ V)
156	139, 151, 154, 155	fvmptd2 6996	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓) = (𝑎‘𝑓))
157	156	oveq1d 7423	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓) = ((𝑎‘𝑓)(.r‘𝐿)𝑓))
158	157	mpteq2dva 5205	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)) = (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑓)(.r‘𝐿)𝑓)))
159		fveq2 6879	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑎‘𝑓) = (𝑎‘𝑣))
160		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → 𝑓 = 𝑣)
161	159, 160	oveq12d 7426	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑣 → ((𝑎‘𝑓)(.r‘𝐿)𝑓) = ((𝑎‘𝑣)(.r‘𝐿)𝑣))
162	161	cbvmptv 5216	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑓)(.r‘𝐿)𝑓)) = (𝑣 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))
163	158, 162	eqtrdi 2820	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)) = (𝑣 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))
164	163	oveq2d 7424	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐿 Σg (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑣 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
165	28	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐿 ∈ CMnd)
166	13	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐻 ∈ (SubDRing‘𝐿))
167		eleq1w 2852	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → (𝑔 ∈ 𝐵 ↔ 𝑓 ∈ 𝐵))
168	167, 149	ifbieq1d 4514	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) = if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)))
169		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿))))
170	169	eldifad 3925	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ 𝐻)
171		fvexd 6894	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (𝑎‘𝑓) ∈ V)
172		fvexd 6894	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (0g‘𝐿) ∈ V)
173	171, 172	ifcld 4536	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)) ∈ V)
174	139, 168, 170, 173	fvmptd3 7011	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓) = if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)))
175	174	oveq1d 7423	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓) = (if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿))(.r‘𝐿)𝑓))
176	126	ad3antrrr 742	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑎 Fn 𝐵)
177	40	ad4antr 744	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
178		fvexd 6894	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → (0g‘𝐿) ∈ V)
179		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵)
180		simplr 780	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿))))
181	180	eldifbd 3926	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → ¬ 𝑓 ∈ (𝑎 supp (0g‘𝐿)))
182	179, 181	eldifd 3924	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿))))
183	176, 177, 178, 182	fvdifsupp 8163	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → (𝑎‘𝑓) = (0g‘𝐿))
184		eqidd 2770	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ ¬ 𝑓 ∈ 𝐵) → (0g‘𝐿) = (0g‘𝐿))
185	183, 184	ifeqda 4526	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)) = (0g‘𝐿))
186	185	oveq1d 7423	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿))(.r‘𝐿)𝑓) = ((0g‘𝐿)(.r‘𝐿)𝑓))
187	27	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝐿 ∈ Ring)
188	166, 14, 63	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐻 ⊆ (Base‘𝐿))
189	188	ssdifssd 4109	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐻 ∖ (𝑎 supp (0g‘𝐿))) ⊆ (Base‘𝐿))
190	189	sselda 3945	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ (Base‘𝐿))
191	4, 5, 6, 187, 190	ringlzd 20374	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → ((0g‘𝐿)(.r‘𝐿)𝑓) = (0g‘𝐿))
192	175, 186, 191	3eqtrd 2808	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓) = (0g‘𝐿))
193		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝑎 finSupp (0g‘𝐿))
194	193	fsuppimpd 9325	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ∈ Fin)
195	27	ad3antrrr 742	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → 𝐿 ∈ Ring)
196	18	ad4antr 744	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → 𝐺 ⊆ (Base‘𝐿))
197	116	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) → 𝑎:𝐵⟶𝐺)
198	197	ffvelcdmda 7077	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ 𝐺)
199	196, 198	sseldd 3946	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ (Base‘𝐿))
200	18, 33	sseldd 3946	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0g‘𝐿) ∈ (Base‘𝐿))
201	200	ad4antr 744	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ ¬ 𝑔 ∈ 𝐵) → (0g‘𝐿) ∈ (Base‘𝐿))
202	199, 201	ifclda 4525	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) ∈ (Base‘𝐿))
203	202	fmpttd 7108	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))):𝐻⟶(Base‘𝐿))
204	203	ffvelcdmda 7077	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓) ∈ (Base‘𝐿))
205	188	sselda 3945	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ (Base‘𝐿))
206	4, 5, 195, 204, 205	ringcld 20338	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓) ∈ (Base‘𝐿))
207	4, 6, 165, 166, 192, 194, 206, 153	gsummptres2 33310	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))
208	113	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
209	126	ad2antrr 738	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑎 Fn 𝐵)
210	208	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
211		fvexd 6894	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → (0g‘𝐿) ∈ V)
212		simpr 489	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿))))
213	209, 210, 211, 212	fvdifsupp 8163	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → (𝑎‘𝑣) = (0g‘𝐿))
214	213	oveq1d 7423	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → ((𝑎‘𝑣)(.r‘𝐿)𝑣) = ((0g‘𝐿)(.r‘𝐿)𝑣))
215	27	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝐿 ∈ Ring)
216	76	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐵 ⊆ (Base‘𝐿))
217	216	ssdifssd 4109	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐵 ∖ (𝑎 supp (0g‘𝐿))) ⊆ (Base‘𝐿))
218	217	sselda 3945	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑣 ∈ (Base‘𝐿))
219	4, 5, 6, 215, 218	ringlzd 20374	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → ((0g‘𝐿)(.r‘𝐿)𝑣) = (0g‘𝐿))
220	214, 219	eqtrd 2804	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → ((𝑎‘𝑣)(.r‘𝐿)𝑣) = (0g‘𝐿))
221	27	ad3antrrr 742	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝐿 ∈ Ring)
222	18	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝐺 ⊆ (Base‘𝐿))
223	116	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝑎:𝐵⟶𝐺)
224	223	ffvelcdmda 7077	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → (𝑎‘𝑣) ∈ 𝐺)
225	222, 224	sseldd 3946	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → (𝑎‘𝑣) ∈ (Base‘𝐿))
226	216	sselda 3945	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ (Base‘𝐿))
227	4, 5, 221, 225, 226	ringcld 20338	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → ((𝑎‘𝑣)(.r‘𝐿)𝑣) ∈ (Base‘𝐿))
228	4, 6, 165, 208, 220, 194, 227, 144	gsummptres2 33310	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))) = (𝐿 Σg (𝑣 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
229	164, 207, 228	3eqtr4d 2814	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))
230	229	eqeq2d 2780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))) ↔ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))))
231	230	biimpar 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)))) → 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))
232	231	anasss 471	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓))))
233	138, 232	jca 520	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → ((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ (((𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))‘𝑓)(.r‘𝐿)𝑓)))))
234	110, 122, 233	rspcedvdw 3593	. . . . . 6 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → ∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))))
235	234	r19.29an 3175	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → ∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))))
236	103, 235	impbida 812	. . . 4 ⊢ (𝜑 → (∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))) ↔ ∃𝑎 ∈ (𝐺 ↑m 𝐵)(𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))))
237	52, 79, 236	3bitr4rd 315	. . 3 ⊢ (𝜑 → (∃𝑝 ∈ (𝐺 ↑m 𝐻)(𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)))) ↔ 𝑥 ∈ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵)))
238	3, 16, 237	3bitrd 308	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵)))
239	238	eqrdv 2767	1 ⊢ (𝜑 → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463   ∖ cdif 3910   ∪ cun 3911   ⊆ wss 3913  ifcif 4489   class class class wbr 5110   ↦ cmpt 5193  dom cdm 5659   Fn wfn 6529  ⟶wf 6530  ‘cfv 6534  (class class class)co 7408   supp csupp 8152   ↑_m cmap 8820  Fincfn 8939   finSupp cfsupp 9317  Basecbs 17265   ↾_s cress 17286  ._rcmulr 17307  Scalarcsca 17309   ·_𝑠 cvsca 17310  0_gc0g 17488   Σ_g cgsu 17489  Mndcmnd 18788  SubGrpcsubg 19182  CMndccmn 19846  Ringcrg 20311  SubRingcsubrg 20650  RingSpancrgspn 20691  Fieldcfield 20810  SubDRingcsdrg 20863  LModclmod 20955  LSpanclspn 21066  LBasisclbs 21169  subringAlg csra 21266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-reg 9550  ax-inf2 9606  ax-ac2 10443  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173  ax-pre-sup 11174  ax-addf 11175

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-isom 6543  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-tpos 8218  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-2o 8450  df-er 8690  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-sup 9398  df-oi 9468  df-r1 9732  df-rank 9733  df-card 9921  df-ac 10096  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-div 11868  df-ind 12215  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-xnn0 12574  df-z 12588  df-dec 12708  df-uz 12859  df-rp 13013  df-fz 13532  df-fzo 13679  df-seq 14034  df-exp 14094  df-hash 14363  df-word 14547  df-lsw 14596  df-concat 14604  df-s1 14630  df-substr 14675  df-pfx 14705  df-s2 14881  df-cj 15146  df-re 15147  df-im 15148  df-sqrt 15282  df-abs 15283  df-clim 15535  df-sum 15734  df-struct 17203  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-ress 17287  df-plusg 17319  df-mulr 17320  df-starv 17321  df-sca 17322  df-vsca 17323  df-ip 17324  df-tset 17325  df-ple 17326  df-ds 17328  df-unif 17329  df-hom 17330  df-cco 17331  df-0g 17490  df-gsum 17491  df-prds 17496  df-pws 17498  df-mre 17634  df-mrc 17635  df-acs 17637  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-mhm 18837  df-submnd 18838  df-grp 18999  df-minusg 19000  df-sbg 19001  df-mulg 19130  df-subg 19185  df-ghm 19280  df-cntz 19383  df-cmn 19848  df-abl 19849  df-mgp 20213  df-rng 20227  df-ur 20260  df-ring 20313  df-cring 20314  df-oppr 20415  df-nzr 20592  df-subrng 20627  df-subrg 20651  df-rgspn 20692  df-drng 20811  df-field 20812  df-sdrg 20864  df-lmod 20957  df-lss 21027  df-lsp 21067  df-lmhm 21117  df-lbs 21170  df-sra 21268  df-rgmod 21269  df-cnfld 21488  df-zring 21562  df-dsmm 21847  df-frlm 21862  df-uvc 21898

This theorem is referenced by:  fldextrspunlem1  34006