Theorem fldextrspunlsp 34064

Description: Lemma for fldextrspunfld 34066. 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 2849	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ (𝑁‘(𝐺 ∪ 𝐻))))
4		eqid 2763	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
5		eqid 2763	. . . 4 ⊢ (.r‘𝐿) = (.r‘𝐿)
6		eqid 2763	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
7		fldextrspunlsp.n	. . . 4 ⊢ 𝑁 = (RingSpan‘𝐿)
8		fldextrspunfld.2	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Field)
9	8	fldcrngd 20842	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
10		fldextrspunfld.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
11		sdrgsubrg 20894	. . . . 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 20894	. . . . 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 33569	. . 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 20671	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
18	12, 17	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
19		eqid 2763	. . . . . . . . 9 ⊢ (𝐿 ↾s 𝐺) = (𝐿 ↾s 𝐺)
20	19, 4	ressbas2 17293	. . . . . . . 8 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘(𝐿 ↾s 𝐺)))
21	18, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (Base‘(𝐿 ↾s 𝐺)))
22		eqidd 2764	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺))
23	22, 18	srasca 21301	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ↾s 𝐺) = (Scalar‘((subringAlg ‘𝐿)‘𝐺)))
24	23	fveq2d 6885	. . . . . . 7 ⊢ (𝜑 → (Base‘(𝐿 ↾s 𝐺)) = (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))))
25	21, 24	eqtr2d 2799	. . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = 𝐺)
26	25	oveq1d 7425	. . . . 5 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↑m 𝐵) = (𝐺 ↑m 𝐵))
27	9	crngringd 20323	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ Ring)
28	27	ringcmnd 20363	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ CMnd)
29	28	cmnmndd 19869	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Mnd)
30		subrgsubg 20676	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ∈ (SubGrp‘𝐿))
31	12, 30	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝐿))
32	6	subg0cl 19195	. . . . . . . . . 10 ⊢ (𝐺 ∈ (SubGrp‘𝐿) → (0g‘𝐿) ∈ 𝐺)
33	31, 32	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐿) ∈ 𝐺)
34	19, 4, 6	ress0g 18815	. . . . . . . . 9 ⊢ ((𝐿 ∈ Mnd ∧ (0g‘𝐿) ∈ 𝐺 ∧ 𝐺 ⊆ (Base‘𝐿)) → (0g‘𝐿) = (0g‘(𝐿 ↾s 𝐺)))
35	29, 33, 18, 34	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝐿) = (0g‘(𝐿 ↾s 𝐺)))
36	23	fveq2d 6885	. . . . . . . 8 ⊢ (𝜑 → (0g‘(𝐿 ↾s 𝐺)) = (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))))
37	35, 36	eqtr2d 2799	. . . . . . 7 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (0g‘𝐿))
38	37	breq2d 5121	. . . . . 6 ⊢ (𝜑 → (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↔ 𝑎 finSupp (0g‘𝐿)))
39		eqid 2763	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺)
40		fldextrspunlsp.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
41	40	mptexd 7222	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)) ∈ V)
42	39	sralmod 21308	. . . . . . . . . 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 33367	. . . . . . . 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 21302	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺)))
46	45	oveqd 7427	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑎‘𝑣)(.r‘𝐿)𝑣) = ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))
47	46	mpteq2dv 5205	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)) = (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣)))
48	47	oveq2d 7426	. . . . . . . 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 2799	. . . . . . 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 2774	. . . . . 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 3339	. . . 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 2763	. . . . 5 ⊢ (LSpan‘((subringAlg ‘𝐿)‘𝐺)) = (LSpan‘((subringAlg ‘𝐿)‘𝐺))
54		eqid 2763	. . . . 5 ⊢ (Base‘((subringAlg ‘𝐿)‘𝐺)) = (Base‘((subringAlg ‘𝐿)‘𝐺))
55		eqid 2763	. . . . 5 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺)))
56		eqid 2763	. . . . 5 ⊢ (Scalar‘((subringAlg ‘𝐿)‘𝐺)) = (Scalar‘((subringAlg ‘𝐿)‘𝐺))
57		eqid 2763	. . . . 5 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺)))
58		eqid 2763	. . . . 5 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺)) = ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))
59		eqid 2763	. . . . . . . . . 10 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
60		eqid 2763	. . . . . . . . . 10 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
61	59, 60	lbsss 21198	. . . . . . . . 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 20671	. . . . . . . . . . 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 17293	. . . . . . . . . 10 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
67	64, 66	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
68		eqidd 2764	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
69		fldextrspunfld.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
70		eqid 2763	. . . . . . . . . . . 12 ⊢ (Base‘𝐽) = (Base‘𝐽)
71	70	sdrgss 20896	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
72	69, 71	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
73	68, 72	srabase 21298	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
74	67, 73	eqtrd 2798	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
75	62, 74	sseqtrrd 3974	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
76	75, 64	sstrd 3947	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝐿))
77	22, 18	srabase 21298	. . . . . 6 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐺)))
78	76, 77	sseqtrd 3973	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((subringAlg ‘𝐿)‘𝐺)))
79	53, 54, 55, 56, 57, 58, 43, 78	ellspds 33683	. . . 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 33025	. . . . . . 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 6881	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑝‘𝑓) = (𝑝‘ℎ))
97		id 23	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
98	96, 97	oveq12d 7428	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑝‘𝑓)(.r‘𝐿)𝑓) = ((𝑝‘ℎ)(.r‘𝐿)ℎ))
99	98	cbvmptv 5215	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)) = (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ))
100	99	oveq2i 7421	. . . . . . . 8 ⊢ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ)))
101	95, 100	eqtrdi 2814	. . . . . . 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 34063	. . . . . 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 3169	. . . . 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 5112	. . . . . . . 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 6880	. . . . . . . . . . . 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 7425	. . . . . . . . . . 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 5205	. . . . . . . . . 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 7426	. . . . . . . . 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 2774	. . . . . . . 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 33025	. . . . . . . . . . . 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 7079	. . . . . . . . . 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 4523	. . . . . . . . 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 7110	. . . . . . . 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 6896	. . . . . . . . 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 6710	. . . . . . . . 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 6706	. . . . . . . . . . . . 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 6896	. . . . . . . . . . . 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 3918	. . . . . . . . . . . . 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 3916	. . . . . . . . . . . 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 2764	. . . . . . . . . . 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 4524	. . . . . . . . . 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 2763	. . . . . . . . . . . . . . . . 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 6716	. . . . . . . . . . . . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ⊆ 𝐵)
145	144	sselda 3937	. . . . . . . . . . . . . . . . . . . . 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 2863	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑔 ∈ 𝐵)
148	147	iftrued 4495	. . . . . . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ⊆ 𝐻)
154	153	sselda 3937	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → 𝑓 ∈ 𝐻)
155		fvexd 6896	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → (𝑎‘𝑓) ∈ V)
156	139, 151, 154, 155	fvmptd2 6998	. . . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . . 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 5204	. . . . . . . . . . . . . 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 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑎‘𝑓) = (𝑎‘𝑣))
160		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → 𝑓 = 𝑣)
161	159, 160	oveq12d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑣 → ((𝑎‘𝑓)(.r‘𝐿)𝑓) = ((𝑎‘𝑣)(.r‘𝐿)𝑣))
162	161	cbvmptv 5215	. . . . . . . . . . . . . 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 2814	. . . . . . . . . . . . 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 7426	. . . . . . . . . . . 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 2846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → (𝑔 ∈ 𝐵 ↔ 𝑓 ∈ 𝐵))
168	167, 149	ifbieq1d 4512	. . . . . . . . . . . . . . . 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 3917	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ 𝐻)
171		fvexd 6896	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (𝑎‘𝑓) ∈ V)
172		fvexd 6896	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (0g‘𝐿) ∈ V)
173	171, 172	ifcld 4534	. . . . . . . . . . . . . . . 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 7013	. . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . 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 4524	. . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . 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 4101	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐻 ∖ (𝑎 supp (0g‘𝐿))) ⊆ (Base‘𝐿))
190	189	sselda 3937	. . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . 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 7079	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ 𝐺)
199	196, 198	sseldd 3938	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ (Base‘𝐿))
200	18, 33	sseldd 3938	. . . . . . . . . . . . . . . . . 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 4523	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) ∈ (Base‘𝐿))
203	202	fmpttd 7110	. . . . . . . . . . . . . . 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 7079	. . . . . . . . . . . . . 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 3937	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ (Base‘𝐿))
206	4, 5, 195, 204, 205	ringcld 20334	. . . . . . . . . . . . 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 33373	. . . . . . . . . . . 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 6896	. . . . . . . . . . . . . . . 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 7425	. . . . . . . . . . . . . 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 4101	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐵 ∖ (𝑎 supp (0g‘𝐿))) ⊆ (Base‘𝐿))
218	217	sselda 3937	. . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . 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 7079	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → (𝑎‘𝑣) ∈ 𝐺)
225	222, 224	sseldd 3938	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → (𝑎‘𝑣) ∈ (Base‘𝐿))
226	216	sselda 3937	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ (Base‘𝐿))
227	4, 5, 221, 225, 226	ringcld 20334	. . . . . . . . . . . . 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 33373	. . . . . . . . . . . 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 2808	. . . . . . . . . . 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 2774	. . . . . . . . . 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 3584	. . . . . 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 3169	. . . . 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 2761	1 ⊢ (𝜑 → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ⊆ wss 3905  ifcif 4487   class class class wbr 5109   ↦ cmpt 5192  dom cdm 5661   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   supp csupp 8152   ↑_m cmap 8820  Fincfn 8939   finSupp cfsupp 9317  Basecbs 17264   ↾_s cress 17285  ._rcmulr 17306  Scalarcsca 17308   ·_𝑠 cvsca 17309  0_gc0g 17487   Σ_g cgsu 17488  Mndcmnd 18787  SubGrpcsubg 19181  CMndccmn 19845  Ringcrg 20310  SubRingcsubrg 20668  RingSpancrgspn 20709  Fieldcfield 20828  SubDRingcsdrg 20889  LModclmod 20981  LSpanclspn 21092  LBasisclbs 21195  subringAlg csra 21292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-reg 9550  ax-inf2 9606  ax-ac2 10442  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173  ax-addf 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-se 5615  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-of 7674  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 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-ind 12214  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-xnn0 12573  df-z 12587  df-dec 12707  df-uz 12858  df-rp 13012  df-fz 13531  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 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-mulr 17319  df-starv 17320  df-sca 17321  df-vsca 17322  df-ip 17323  df-tset 17324  df-ple 17325  df-ds 17327  df-unif 17328  df-hom 17329  df-cco 17330  df-0g 17489  df-gsum 17490  df-prds 17495  df-pws 17497  df-mre 17633  df-mrc 17634  df-acs 17636  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-mhm 18836  df-submnd 18837  df-grp 18998  df-minusg 18999  df-sbg 19000  df-mulg 19129  df-subg 19184  df-ghm 19279  df-cntz 19382  df-cmn 19847  df-abl 19848  df-mgp 20212  df-rng 20226  df-ur 20259  df-ring 20312  df-cring 20313  df-oppr 20415  df-nzr 20610  df-subrng 20645  df-subrg 20669  df-rgspn 20710  df-drng 20829  df-field 20830  df-sdrg 20890  df-lmod 20983  df-lss 21053  df-lsp 21093  df-lmhm 21143  df-lbs 21196  df-sra 21294  df-rgmod 21295  df-cnfld 21523  df-zring 21597  df-dsmm 21882  df-frlm 21897  df-uvc 21933

This theorem is referenced by:  fldextrspunlem1  34065