Theorem fldextrspunlsp 34184

Description: Lemma for fldextrspunfld 34186. 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 2846	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ (𝑁‘(𝐺 ∪ 𝐻))))
4		eqid 2760	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
5		eqid 2760	. . . 4 ⊢ (.r‘𝐿) = (.r‘𝐿)
6		eqid 2760	. . . 4 ⊢ (0g‘𝐿) = (0g‘𝐿)
7		fldextrspunlsp.n	. . . 4 ⊢ 𝑁 = (RingSpan‘𝐿)
8		fldextrspunfld.2	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ Field)
9	8	fldcrngd 20905	. . . 4 ⊢ (𝜑 → 𝐿 ∈ CRing)
10		fldextrspunfld.5	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (SubDRing‘𝐿))
11		sdrgsubrg 20957	. . . . 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 20957	. . . . 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 33689	. . 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 20734	. . . . . . . . 9 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ⊆ (Base‘𝐿))
18	12, 17	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ⊆ (Base‘𝐿))
19		eqid 2760	. . . . . . . . 9 ⊢ (𝐿 ↾s 𝐺) = (𝐿 ↾s 𝐺)
20	19, 4	ressbas2 17330	. . . . . . . 8 ⊢ (𝐺 ⊆ (Base‘𝐿) → 𝐺 = (Base‘(𝐿 ↾s 𝐺)))
21	18, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐺 = (Base‘(𝐿 ↾s 𝐺)))
22		eqidd 2761	. . . . . . . . 9 ⊢ (𝜑 → ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺))
23	22, 18	srasca 21364	. . . . . . . 8 ⊢ (𝜑 → (𝐿 ↾s 𝐺) = (Scalar‘((subringAlg ‘𝐿)‘𝐺)))
24	23	fveq2d 6882	. . . . . . 7 ⊢ (𝜑 → (Base‘(𝐿 ↾s 𝐺)) = (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))))
25	21, 24	eqtr2d 2796	. . . . . 6 ⊢ (𝜑 → (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = 𝐺)
26	25	oveq1d 7428	. . . . 5 ⊢ (𝜑 → ((Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↑m 𝐵) = (𝐺 ↑m 𝐵))
27	9	crngringd 20385	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐿 ∈ Ring)
28	27	ringcmnd 20425	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ CMnd)
29	28	cmnmndd 19931	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ Mnd)
30		subrgsubg 20739	. . . . . . . . . . 11 ⊢ (𝐺 ∈ (SubRing‘𝐿) → 𝐺 ∈ (SubGrp‘𝐿))
31	12, 30	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 ∈ (SubGrp‘𝐿))
32	6	subg0cl 19257	. . . . . . . . . 10 ⊢ (𝐺 ∈ (SubGrp‘𝐿) → (0g‘𝐿) ∈ 𝐺)
33	31, 32	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐿) ∈ 𝐺)
34	19, 4, 6	ress0g 18867	. . . . . . . . 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 6882	. . . . . . . 8 ⊢ (𝜑 → (0g‘(𝐿 ↾s 𝐺)) = (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))))
37	35, 36	eqtr2d 2796	. . . . . . 7 ⊢ (𝜑 → (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (0g‘𝐿))
38	37	breq2d 5115	. . . . . 6 ⊢ (𝜑 → (𝑎 finSupp (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) ↔ 𝑎 finSupp (0g‘𝐿)))
39		eqid 2760	. . . . . . . . 9 ⊢ ((subringAlg ‘𝐿)‘𝐺) = ((subringAlg ‘𝐿)‘𝐺)
40		fldextrspunlsp.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
41	40	mptexd 7223	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)) ∈ V)
42	39	sralmod 21371	. . . . . . . . . 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 33487	. . . . . . . 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 21365	. . . . . . . . . . 11 ⊢ (𝜑 → (.r‘𝐿) = ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺)))
46	45	oveqd 7430	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑎‘𝑣)(.r‘𝐿)𝑣) = ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣))
47	46	mpteq2dv 5199	. . . . . . . . 9 ⊢ (𝜑 → (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣)) = (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))𝑣)))
48	47	oveq2d 7429	. . . . . . . 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 2796	. . . . . . 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 2771	. . . . . 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 644	. . . . 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 3335	. . . 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 2760	. . . . 5 ⊢ (LSpan‘((subringAlg ‘𝐿)‘𝐺)) = (LSpan‘((subringAlg ‘𝐿)‘𝐺))
54		eqid 2760	. . . . 5 ⊢ (Base‘((subringAlg ‘𝐿)‘𝐺)) = (Base‘((subringAlg ‘𝐿)‘𝐺))
55		eqid 2760	. . . . 5 ⊢ (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (Base‘(Scalar‘((subringAlg ‘𝐿)‘𝐺)))
56		eqid 2760	. . . . 5 ⊢ (Scalar‘((subringAlg ‘𝐿)‘𝐺)) = (Scalar‘((subringAlg ‘𝐿)‘𝐺))
57		eqid 2760	. . . . 5 ⊢ (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺))) = (0g‘(Scalar‘((subringAlg ‘𝐿)‘𝐺)))
58		eqid 2760	. . . . 5 ⊢ ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺)) = ( ·𝑠 ‘((subringAlg ‘𝐿)‘𝐺))
59		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘((subringAlg ‘𝐽)‘𝐹)) = (Base‘((subringAlg ‘𝐽)‘𝐹))
60		eqid 2760	. . . . . . . . . 10 ⊢ (LBasis‘((subringAlg ‘𝐽)‘𝐹)) = (LBasis‘((subringAlg ‘𝐽)‘𝐹))
61	59, 60	lbsss 21261	. . . . . . . . 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 20734	. . . . . . . . . . 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 17330	. . . . . . . . . 10 ⊢ (𝐻 ⊆ (Base‘𝐿) → 𝐻 = (Base‘𝐽))
67	64, 66	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐻 = (Base‘𝐽))
68		eqidd 2761	. . . . . . . . . 10 ⊢ (𝜑 → ((subringAlg ‘𝐽)‘𝐹) = ((subringAlg ‘𝐽)‘𝐹))
69		fldextrspunfld.4	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (SubDRing‘𝐽))
70		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝐽) = (Base‘𝐽)
71	70	sdrgss 20959	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (SubDRing‘𝐽) → 𝐹 ⊆ (Base‘𝐽))
72	69, 71	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ⊆ (Base‘𝐽))
73	68, 72	srabase 21361	. . . . . . . . 9 ⊢ (𝜑 → (Base‘𝐽) = (Base‘((subringAlg ‘𝐽)‘𝐹)))
74	67, 73	eqtrd 2795	. . . . . . . 8 ⊢ (𝜑 → 𝐻 = (Base‘((subringAlg ‘𝐽)‘𝐹)))
75	62, 74	sseqtrrd 3968	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐻)
76	75, 64	sstrd 3941	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ (Base‘𝐿))
77	22, 18	srabase 21361	. . . . . 6 ⊢ (𝜑 → (Base‘𝐿) = (Base‘((subringAlg ‘𝐿)‘𝐺)))
78	76, 77	sseqtrd 3967	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ (Base‘((subringAlg ‘𝐿)‘𝐺)))
79	53, 54, 55, 56, 57, 58, 43, 78	ellspds 33803	. . . 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 739	. . . . . . 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 739	. . . . . . 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 739	. . . . . . 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 739	. . . . . . 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 739	. . . . . . 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 739	. . . . . . 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 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑝 ∈ (𝐺 ↑m 𝐻)) ∧ (𝑝 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))))) → 𝐵 ∈ Fin)
92		simplr 781	. . . . . . . 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	92	elmaprd 8849	. . . . . . 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 783	. . . . . . 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 785	. . . . . . . 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 6878	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → (𝑝‘𝑓) = (𝑝‘ℎ))
97		id 23	. . . . . . . . . . 11 ⊢ (𝑓 = ℎ → 𝑓 = ℎ)
98	96, 97	oveq12d 7431	. . . . . . . . . 10 ⊢ (𝑓 = ℎ → ((𝑝‘𝑓)(.r‘𝐿)𝑓) = ((𝑝‘ℎ)(.r‘𝐿)ℎ))
99	98	cbvmptv 5209	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓)) = (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ))
100	99	oveq2i 7424	. . . . . . . 8 ⊢ (𝐿 Σg (𝑓 ∈ 𝐻 ↦ ((𝑝‘𝑓)(.r‘𝐿)𝑓))) = (𝐿 Σg (ℎ ∈ 𝐻 ↦ ((𝑝‘ℎ)(.r‘𝐿)ℎ)))
101	95, 100	eqtrdi 2811	. . . . . . 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 34183	. . . . . 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 3166	. . . . 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 5106	. . . . . . . 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 6877	. . . . . . . . . . . 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 7428	. . . . . . . . . . 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 5199	. . . . . . . . . 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 7429	. . . . . . . . 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 2771	. . . . . . . 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 644	. . . . . . 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 739	. . . . . . . 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 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝐻 ∈ (SubDRing‘𝐿))
113		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝑎 ∈ (𝐺 ↑m 𝐵))
114	113	elmaprd 8849	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝑎:𝐵⟶𝐺)
115	114	ad2antrr 739	. . . . . . . . . . 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)
116	115	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)
117	33	ad4antr 745	. . . . . . . . . 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)
118	116, 117	ifclda 4518	. . . . . . . . 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)
119	118	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)
120	111, 112, 119	elmapdd 8840	. . . . . . 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))
121		fvexd 6893	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → (0g‘𝐿) ∈ V)
122	119	ffund 6707	. . . . . . . . 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))))
123		simprl 783	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ (𝑎 finSupp (0g‘𝐿) ∧ 𝑥 = (𝐿 Σg (𝑣 ∈ 𝐵 ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))))) → 𝑎 finSupp (0g‘𝐿))
124	114	ffnd 6703	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝑎 Fn 𝐵)
125	124	ad3antrrr 743	. . . . . . . . . . . 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)
126	40	ad4antr 745	. . . . . . . . . . . 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)))
127		fvexd 6893	. . . . . . . . . . . 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)
128		simpr 490	. . . . . . . . . . . . 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)
129		simplr 781	. . . . . . . . . . . . . 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))))
130	129	eldifbd 3912	. . . . . . . . . . . . 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)))
131	128, 130	eldifd 3910	. . . . . . . . . . . 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))))
132	125, 126, 127, 131	fvdifsupp 8169	. . . . . . . . . . 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))
133		eqidd 2761	. . . . . . . . . . 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))
134	132, 133	ifeqda 4519	. . . . . . . . . 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))
135	134, 112	suppss2 8198	. . . . . . . . 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)))
136	120, 121, 122, 123, 135	fsuppsssuppgd 9352	. . . . . . . 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))
137		eqid 2760	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))) = (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)))
138		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑔 = 𝑓)
139		suppssdm 8175	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 supp (0g‘𝐿)) ⊆ dom 𝑎
140	114	fdmd 6713	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → dom 𝑎 = 𝐵)
141	140	adantr 486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → dom 𝑎 = 𝐵)
142	139, 141	sseqtrid 3973	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ⊆ 𝐵)
143	142	sselda 3931	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → 𝑓 ∈ 𝐵)
144	143	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑓 ∈ 𝐵)
145	138, 144	eqeltrd 2860	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → 𝑔 ∈ 𝐵)
146	145	iftrued 4490	. . . . . . . . . . . . . . . . . 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))
147		fveq2 6878	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = 𝑓 → (𝑎‘𝑔) = (𝑎‘𝑓))
148	147	adantl 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) ∧ 𝑔 = 𝑓) → (𝑎‘𝑔) = (𝑎‘𝑓))
149	146, 148	eqtrd 2795	. . . . . . . . . . . . . . . . 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))
150	75	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐵 ⊆ 𝐻)
151	142, 150	sstrd 3941	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ⊆ 𝐻)
152	151	sselda 3931	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → 𝑓 ∈ 𝐻)
153		fvexd 6893	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝑎 supp (0g‘𝐿))) → (𝑎‘𝑓) ∈ V)
154	137, 149, 152, 153	fvmptd2 6995	. . . . . . . . . . . . . . . 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))
155	154	oveq1d 7428	. . . . . . . . . . . . . . 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))
156	155	mpteq2dva 5198	. . . . . . . . . . . . . 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)))
157		fveq2 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → (𝑎‘𝑓) = (𝑎‘𝑣))
158		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑣 → 𝑓 = 𝑣)
159	157, 158	oveq12d 7431	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑣 → ((𝑎‘𝑓)(.r‘𝐿)𝑓) = ((𝑎‘𝑣)(.r‘𝐿)𝑣))
160	159	cbvmptv 5209	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑓)(.r‘𝐿)𝑓)) = (𝑣 ∈ (𝑎 supp (0g‘𝐿)) ↦ ((𝑎‘𝑣)(.r‘𝐿)𝑣))
161	156, 160	eqtrdi 2811	. . . . . . . . . . . . 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)))
162	161	oveq2d 7429	. . . . . . . . . . . 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))))
163	28	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐿 ∈ CMnd)
164	13	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐻 ∈ (SubDRing‘𝐿))
165		eleq1w 2843	. . . . . . . . . . . . . . . . 17 ⊢ (𝑔 = 𝑓 → (𝑔 ∈ 𝐵 ↔ 𝑓 ∈ 𝐵))
166	165, 147	ifbieq1d 4507	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 = 𝑓 → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) = if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)))
167		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿))))
168	167	eldifad 3911	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ 𝐻)
169		fvexd 6893	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (𝑎‘𝑓) ∈ V)
170		fvexd 6893	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → (0g‘𝐿) ∈ V)
171	169, 170	ifcld 4529	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → if(𝑓 ∈ 𝐵, (𝑎‘𝑓), (0g‘𝐿)) ∈ V)
172	137, 166, 168, 171	fvmptd3 7010	. . . . . . . . . . . . . . 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)))
173	172	oveq1d 7428	. . . . . . . . . . . . . 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))
174	124	ad3antrrr 743	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑎 Fn 𝐵)
175	40	ad4antr 745	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
176		fvexd 6893	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → (0g‘𝐿) ∈ V)
177		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ 𝐵)
178		simplr 781	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿))))
179	178	eldifbd 3912	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → ¬ 𝑓 ∈ (𝑎 supp (0g‘𝐿)))
180	177, 179	eldifd 3910	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → 𝑓 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿))))
181	174, 175, 176, 180	fvdifsupp 8169	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ 𝑓 ∈ 𝐵) → (𝑎‘𝑓) = (0g‘𝐿))
182		eqidd 2761	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) ∧ ¬ 𝑓 ∈ 𝐵) → (0g‘𝐿) = (0g‘𝐿))
183	181, 182	ifeqda 4519	. . . . . . . . . . . . . . 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))
184	183	oveq1d 7428	. . . . . . . . . . . . . 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))
185	27	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝐿 ∈ Ring)
186	164, 14, 63	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐻 ⊆ (Base‘𝐿))
187	186	ssdifssd 4094	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐻 ∖ (𝑎 supp (0g‘𝐿))) ⊆ (Base‘𝐿))
188	187	sselda 3931	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑓 ∈ (Base‘𝐿))
189	4, 5, 6, 185, 188	ringlzd 20437	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ (𝐻 ∖ (𝑎 supp (0g‘𝐿)))) → ((0g‘𝐿)(.r‘𝐿)𝑓) = (0g‘𝐿))
190	173, 184, 189	3eqtrd 2799	. . . . . . . . . . . . 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))
191		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝑎 finSupp (0g‘𝐿))
192	191	fsuppimpd 9339	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑎 supp (0g‘𝐿)) ∈ Fin)
193	27	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → 𝐿 ∈ Ring)
194	18	ad4antr 745	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → 𝐺 ⊆ (Base‘𝐿))
195	114	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) → 𝑎:𝐵⟶𝐺)
196	195	ffvelcdmda 7077	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ 𝐺)
197	194, 196	sseldd 3932	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ 𝑔 ∈ 𝐵) → (𝑎‘𝑔) ∈ (Base‘𝐿))
198	18, 33	sseldd 3932	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (0g‘𝐿) ∈ (Base‘𝐿))
199	198	ad4antr 745	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) ∧ ¬ 𝑔 ∈ 𝐵) → (0g‘𝐿) ∈ (Base‘𝐿))
200	197, 199	ifclda 4518	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑔 ∈ 𝐻) → if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿)) ∈ (Base‘𝐿))
201	200	fmpttd 7108	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝑔 ∈ 𝐻 ↦ if(𝑔 ∈ 𝐵, (𝑎‘𝑔), (0g‘𝐿))):𝐻⟶(Base‘𝐿))
202	201	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))
203	186	sselda 3931	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑓 ∈ 𝐻) → 𝑓 ∈ (Base‘𝐿))
204	4, 5, 193, 202, 203	ringcld 20396	. . . . . . . . . . . . 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))
205	4, 6, 163, 164, 190, 192, 204, 151	gsummptres2 33493	. . . . . . . . . . . 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))))
206	40	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
207	206	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
208	124	ad2antrr 739	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑎 Fn 𝐵)
209	207	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝐵 ∈ (LBasis‘((subringAlg ‘𝐽)‘𝐹)))
210		fvexd 6893	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → (0g‘𝐿) ∈ V)
211		simpr 490	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿))))
212	208, 209, 210, 211	fvdifsupp 8169	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → (𝑎‘𝑣) = (0g‘𝐿))
213	212	oveq1d 7428	. . . . . . . . . . . . . 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))
214	27	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝐿 ∈ Ring)
215	76	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝐵 ⊆ (Base‘𝐿))
216	215	ssdifssd 4094	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → (𝐵 ∖ (𝑎 supp (0g‘𝐿))) ⊆ (Base‘𝐿))
217	216	sselda 3931	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → 𝑣 ∈ (Base‘𝐿))
218	4, 5, 6, 214, 217	ringlzd 20437	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → ((0g‘𝐿)(.r‘𝐿)𝑣) = (0g‘𝐿))
219	213, 218	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ (𝐵 ∖ (𝑎 supp (0g‘𝐿)))) → ((𝑎‘𝑣)(.r‘𝐿)𝑣) = (0g‘𝐿))
220	27	ad3antrrr 743	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝐿 ∈ Ring)
221	18	ad3antrrr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝐺 ⊆ (Base‘𝐿))
222	114	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) → 𝑎:𝐵⟶𝐺)
223	222	ffvelcdmda 7077	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → (𝑎‘𝑣) ∈ 𝐺)
224	221, 223	sseldd 3932	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → (𝑎‘𝑣) ∈ (Base‘𝐿))
225	215	sselda 3931	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → 𝑣 ∈ (Base‘𝐿))
226	4, 5, 220, 224, 225	ringcld 20396	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ (𝐺 ↑m 𝐵)) ∧ 𝑎 finSupp (0g‘𝐿)) ∧ 𝑣 ∈ 𝐵) → ((𝑎‘𝑣)(.r‘𝐿)𝑣) ∈ (Base‘𝐿))
227	4, 6, 163, 207, 219, 192, 226, 142	gsummptres2 33493	. . . . . . . . . . . 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))))
228	162, 205, 227	3eqtr4d 2805	. . . . . . . . . . 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))))
229	228	eqeq2d 2771	. . . . . . . . . 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)))))
230	229	biimpar 483	. . . . . . . . 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))))
231	230	anasss 472	. . . . . . . 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))))
232	136, 231	jca 521	. . . . . . 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)))))
233	110, 120, 232	rspcedvdw 3579	. . . . . 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)))))
234	233	r19.29an 3166	. . . . 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)))))
235	103, 234	impbida 813	. . . 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))))))
236	52, 79, 235	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)))
237	3, 16, 236	3bitrd 308	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵)))
238	237	eqrdv 2758	1 ⊢ (𝜑 → 𝐶 = ((LSpan‘((subringAlg ‘𝐿)‘𝐺))‘𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  Vcvv 3450   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ifcif 4482   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5655   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   supp csupp 8158   ↑_m cmap 8826  Fincfn 8952   finSupp cfsupp 9331  Basecbs 17301   ↾_s cress 17322  ._rcmulr 17343  Scalarcsca 17345   ·_𝑠 cvsca 17346  0_gc0g 17524   Σ_g cgsu 17525  Mndcmnd 18836  SubGrpcsubg 19243  CMndccmn 19907  Ringcrg 20372  SubRingcsubrg 20731  RingSpancrgspn 20772  Fieldcfield 20891  SubDRingcsdrg 20952  LModclmod 21044  LSpanclspn 21155  LBasisclbs 21258  subringAlg csra 21355

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-reg 9564  ax-inf2 9620  ax-ac2 10465  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202  ax-addf 11203

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8159  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-ixp 8905  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-fsupp 9332  df-sup 9412  df-oi 9482  df-r1 9746  df-rank 9747  df-scott 9868  df-card 9944  df-ac 10119  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-ind 12243  df-nn 12258  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12737  df-uz 12888  df-rp 13043  df-fz 13562  df-fzo 13710  df-seq 14066  df-exp 14126  df-hash 14395  df-word 14579  df-lsw 14628  df-concat 14636  df-s1 14663  df-substr 14709  df-pfx 14741  df-s2 14919  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-sum 15774  df-struct 17239  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-mulr 17356  df-starv 17357  df-sca 17358  df-vsca 17359  df-ip 17360  df-tset 17361  df-ple 17362  df-ds 17364  df-unif 17365  df-hom 17366  df-cco 17367  df-0g 17526  df-gsum 17527  df-prds 17532  df-pws 17534  df-mre 17670  df-mrc 17671  df-acs 17673  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-mhm 18891  df-submnd 18892  df-grp 19060  df-minusg 19061  df-sbg 19062  df-mulg 19191  df-subg 19246  df-ghm 19341  df-cntz 19444  df-cmn 19909  df-abl 19910  df-mgp 20274  df-rng 20288  df-ur 20321  df-ring 20374  df-cring 20375  df-oppr 20478  df-nzr 20673  df-subrng 20708  df-subrg 20732  df-rgspn 20773  df-drng 20892  df-field 20893  df-sdrg 20953  df-lmod 21046  df-lss 21116  df-lsp 21156  df-lmhm 21206  df-lbs 21259  df-sra 21357  df-rgmod 21358  df-cnfld 21586  df-zring 21660  df-dsmm 21945  df-frlm 21960  df-uvc 21996

This theorem is used by:  fldextrspunlem1  34185