Theorem gsumvsca1 33195

Description: Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.)

Hypotheses

Ref	Expression
gsumvsca.b	⊢ 𝐵 = (Base‘𝑊)
gsumvsca.g	⊢ 𝐺 = (Scalar‘𝑊)
gsumvsca.z	⊢ 0 = (0g‘𝑊)
gsumvsca.t	⊢ · = ( ·𝑠 ‘𝑊)
gsumvsca.p	⊢ + = (+g‘𝑊)
gsumvsca.k	⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺))
gsumvsca.a	⊢ (𝜑 → 𝐴 ∈ Fin)
gsumvsca.w	⊢ (𝜑 → 𝑊 ∈ SLMod)
gsumvsca1.n	⊢ (𝜑 → 𝑃 ∈ 𝐾)
gsumvsca1.c	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑄 ∈ 𝐵)

Assertion

Ref	Expression
gsumvsca1	⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))

Distinct variable groups:   · ,𝑘   𝐴,𝑘   𝑘,𝑊   𝜑,𝑘   𝐵,𝑘   𝑃,𝑘

Allowed substitution hints:   + (𝑘)   𝑄(𝑘)   𝐺(𝑘)   𝐾(𝑘)   0 (𝑘)

Proof of Theorem gsumvsca1

Dummy variables 𝑒 𝑎 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumvsca.a	. 2 ⊢ (𝜑 → 𝐴 ∈ Fin)
2		ssid 3966	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		sseq1 3969	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4	3	anbi2d 630	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
5		mpteq1 5191	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄)))
6	5	oveq2d 7385	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))))
7		mpteq1 5191	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ ∅ ↦ 𝑄))
8	7	oveq2d 7385	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)))
9	8	oveq2d 7385	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))))
10	6, 9	eqeq12d 2745	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) ↔ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)))))
11	4, 10	imbi12d 344	. . . . 5 ⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)))) ↔ ((𝜑 ∧ ∅ ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))))))
12		sseq1 3969	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑎 ⊆ 𝐴 ↔ 𝑒 ⊆ 𝐴))
13	12	anbi2d 630	. . . . . 6 ⊢ (𝑎 = 𝑒 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝑒 ⊆ 𝐴)))
14		mpteq1 5191	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄)))
15	14	oveq2d 7385	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))))
16		mpteq1 5191	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ 𝑒 ↦ 𝑄))
17	16	oveq2d 7385	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))
18	17	oveq2d 7385	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))))
19	15, 18	eqeq12d 2745	. . . . . 6 ⊢ (𝑎 = 𝑒 → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) ↔ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))))
20	13, 19	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝑒 → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)))) ↔ ((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))))))
21		sseq1 3969	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑎 ⊆ 𝐴 ↔ (𝑒 ∪ {𝑧}) ⊆ 𝐴))
22	21	anbi2d 630	. . . . . 6 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)))
23		mpteq1 5191	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄)))
24	23	oveq2d 7385	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))))
25		mpteq1 5191	. . . . . . . . 9 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))
26	25	oveq2d 7385	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄)))
27	26	oveq2d 7385	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))
28	24, 27	eqeq12d 2745	. . . . . 6 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) ↔ (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄)))))
29	22, 28	imbi12d 344	. . . . 5 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)))) ↔ ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))))
30		sseq1 3969	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
31	30	anbi2d 630	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐴)))
32		mpteq1 5191	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄)))
33	32	oveq2d 7385	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))))
34		mpteq1 5191	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ 𝐴 ↦ 𝑄))
35	34	oveq2d 7385	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))
36	35	oveq2d 7385	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))
37	33, 36	eqeq12d 2745	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) ↔ (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))))
38	31, 37	imbi12d 344	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)))) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))))
39		gsumvsca.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ SLMod)
40		gsumvsca.k	. . . . . . . . . 10 ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺))
41		gsumvsca1.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝐾)
42	40, 41	sseldd 3944	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐺))
43		gsumvsca.g	. . . . . . . . . 10 ⊢ 𝐺 = (Scalar‘𝑊)
44		gsumvsca.t	. . . . . . . . . 10 ⊢ · = ( ·𝑠 ‘𝑊)
45		eqid 2729	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
46		gsumvsca.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑊)
47	43, 44, 45, 46	slmdvs0 33194	. . . . . . . . 9 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺)) → (𝑃 · 0 ) = 0 )
48	39, 42, 47	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (𝑃 · 0 ) = 0 )
49	48	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → 0 = (𝑃 · 0 ))
50		mpt0 6642	. . . . . . . . 9 ⊢ (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄)) = ∅
51	50	oveq2i 7380	. . . . . . . 8 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑊 Σg ∅)
52	46	gsum0 18593	. . . . . . . 8 ⊢ (𝑊 Σg ∅) = 0
53	51, 52	eqtri 2752	. . . . . . 7 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = 0
54		mpt0 6642	. . . . . . . . . 10 ⊢ (𝑘 ∈ ∅ ↦ 𝑄) = ∅
55	54	oveq2i 7380	. . . . . . . . 9 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)) = (𝑊 Σg ∅)
56	55, 52	eqtri 2752	. . . . . . . 8 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)) = 0
57	56	oveq2i 7380	. . . . . . 7 ⊢ (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))) = (𝑃 · 0 )
58	49, 53, 57	3eqtr4g 2789	. . . . . 6 ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))))
59	58	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))))
60		ssun1 4137	. . . . . . . . 9 ⊢ 𝑒 ⊆ (𝑒 ∪ {𝑧})
61		sstr2 3950	. . . . . . . . 9 ⊢ (𝑒 ⊆ (𝑒 ∪ {𝑧}) → ((𝑒 ∪ {𝑧}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴))
62	60, 61	ax-mp 5	. . . . . . . 8 ⊢ ((𝑒 ∪ {𝑧}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴)
63	62	anim2i 617	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝜑 ∧ 𝑒 ⊆ 𝐴))
64	63	imim1i 63	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))))
65	39	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑊 ∈ SLMod)
66	42	ad2antrl 728	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑃 ∈ (Base‘𝐺))
67		gsumvsca.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑊)
68		slmdcmn 33174	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
69	65, 68	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑊 ∈ CMnd)
70		vex 3448	. . . . . . . . . . . . 13 ⊢ 𝑒 ∈ V
71	70	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ∈ V)
72		simplrl 776	. . . . . . . . . . . . . 14 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝜑)
73		simprr 772	. . . . . . . . . . . . . . . 16 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑒 ∪ {𝑧}) ⊆ 𝐴)
74	73	unssad 4152	. . . . . . . . . . . . . . 15 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ⊆ 𝐴)
75	74	sselda 3943	. . . . . . . . . . . . . 14 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑘 ∈ 𝐴)
76		gsumvsca1.c	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑄 ∈ 𝐵)
77	72, 75, 76	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑄 ∈ 𝐵)
78	77	fmpttd 7069	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑘 ∈ 𝑒 ↦ 𝑄):𝑒⟶𝐵)
79		eqid 2729	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑒 ↦ 𝑄) = (𝑘 ∈ 𝑒 ↦ 𝑄)
80		simpll 766	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ∈ Fin)
81	46	fvexi 6854	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
82	81	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 0 ∈ V)
83	79, 80, 77, 82	fsuppmptdm 9303	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑘 ∈ 𝑒 ↦ 𝑄) finSupp 0 )
84	67, 46, 69, 71, 78, 83	gsumcl 19829	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) ∈ 𝐵)
85	73	unssbd 4153	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → {𝑧} ⊆ 𝐴)
86		vex 3448	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
87	86	snss 4745	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
88	85, 87	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ 𝐴)
89	76	ralrimiva 3125	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑄 ∈ 𝐵)
90	89	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ∀𝑘 ∈ 𝐴 𝑄 ∈ 𝐵)
91		rspcsbela 4397	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝑄 ∈ 𝐵) → ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵)
92	88, 90, 91	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵)
93		gsumvsca.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑊)
94	67, 93, 43, 44, 45	slmdvsdi 33184	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ SLMod ∧ (𝑃 ∈ (Base‘𝐺) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) ∈ 𝐵 ∧ ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵)) → (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
95	65, 66, 84, 92, 94	syl13anc 1374	. . . . . . . . . 10 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
96	95	adantr 480	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
97		nfcsb1v 3883	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝑄
98	86	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ V)
99		simplr 768	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧 ∈ 𝑒)
100		csbeq1a 3873	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → 𝑄 = ⦋𝑧 / 𝑘⦌𝑄)
101	97, 67, 93, 69, 80, 77, 98, 99, 92, 100	gsumunsnf 19873	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄)) = ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄))
102	101	oveq2d 7385	. . . . . . . . . 10 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))) = (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)))
103	102	adantr 480	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))) = (𝑃 · ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) + ⦋𝑧 / 𝑘⦌𝑄)))
104		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝑃
105		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 ·
106	104, 105, 97	nfov 7399	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑃 · ⦋𝑧 / 𝑘⦌𝑄)
107	72, 39	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑊 ∈ SLMod)
108	72, 42	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑃 ∈ (Base‘𝐺))
109	67, 43, 44, 45	slmdvscl 33183	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺) ∧ 𝑄 ∈ 𝐵) → (𝑃 · 𝑄) ∈ 𝐵)
110	107, 108, 77, 109	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → (𝑃 · 𝑄) ∈ 𝐵)
111	67, 43, 44, 45	slmdvscl 33183	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺) ∧ ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵) → (𝑃 · ⦋𝑧 / 𝑘⦌𝑄) ∈ 𝐵)
112	65, 66, 92, 111	syl3anc 1373	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑃 · ⦋𝑧 / 𝑘⦌𝑄) ∈ 𝐵)
113	100	oveq2d 7385	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑃 · 𝑄) = (𝑃 · ⦋𝑧 / 𝑘⦌𝑄))
114	106, 67, 93, 69, 80, 110, 98, 99, 112, 113	gsumunsnf 19873	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
115	114	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
116		simpr 484	. . . . . . . . . . 11 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))))
117	116	oveq1d 7384	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
118	115, 117	eqtrd 2764	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) + (𝑃 · ⦋𝑧 / 𝑘⦌𝑄)))
119	96, 103, 118	3eqtr4rd 2775	. . . . . . . 8 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))
120	119	exp31 419	. . . . . . 7 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))))
121	120	a2d 29	. . . . . 6 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) → (((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))))
122	64, 121	syl5 34	. . . . 5 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) → (((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))))
123	11, 20, 29, 38, 59, 122	findcard2s 9106	. . . 4 ⊢ (𝐴 ∈ Fin → ((𝜑 ∧ 𝐴 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))))
124	123	imp 406	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝜑 ∧ 𝐴 ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))
125	2, 124	mpanr2 704	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝜑) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))
126	1, 125	mpancom 688	1 ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3444  ⦋csb 3859   ∪ cun 3909   ⊆ wss 3911  ∅c0 4292  {csn 4585   ↦ cmpt 5183  ‘cfv 6499  (class class class)co 7369  Fincfn 8895  Basecbs 17155  +_gcplusg 17196  Scalarcsca 17199   ·_𝑠 cvsca 17200  0_gc0g 17378   Σ_g cgsu 17379  CMndccmn 19694  SLModcslmd 33169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120  ax-pre-mulgt0 11121

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-of 7633  df-om 7823  df-1st 7947  df-2nd 7948  df-supp 8117  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-fsupp 9289  df-oi 9439  df-card 9868  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189  df-le 11190  df-sub 11383  df-neg 11384  df-nn 12163  df-2 12225  df-n0 12419  df-z 12506  df-uz 12770  df-fz 13445  df-fzo 13592  df-seq 13943  df-hash 14272  df-sets 17110  df-slot 17128  df-ndx 17140  df-base 17156  df-ress 17177  df-plusg 17209  df-0g 17380  df-gsum 17381  df-mre 17523  df-mrc 17524  df-acs 17526  df-mgm 18549  df-sgrp 18628  df-mnd 18644  df-submnd 18693  df-mulg 18982  df-cntz 19231  df-cmn 19696  df-srg 20107  df-slmd 33170

This theorem is referenced by: (None)