Theorem gsumvsca1 33287

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 3955	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		sseq1 3958	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4	3	anbi2d 631	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
5		mpteq1 5186	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄)))
6	5	oveq2d 7374	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))))
7		mpteq1 5186	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ ∅ ↦ 𝑄))
8	7	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)))
9	8	oveq2d 7374	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))))
10	6, 9	eqeq12d 2751	. . . . . 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 3958	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑎 ⊆ 𝐴 ↔ 𝑒 ⊆ 𝐴))
13	12	anbi2d 631	. . . . . 6 ⊢ (𝑎 = 𝑒 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝑒 ⊆ 𝐴)))
14		mpteq1 5186	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄)))
15	14	oveq2d 7374	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))))
16		mpteq1 5186	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ 𝑒 ↦ 𝑄))
17	16	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)))
18	17	oveq2d 7374	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄))))
19	15, 18	eqeq12d 2751	. . . . . 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 3958	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑎 ⊆ 𝐴 ↔ (𝑒 ∪ {𝑧}) ⊆ 𝐴))
22	21	anbi2d 631	. . . . . 6 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)))
23		mpteq1 5186	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄)))
24	23	oveq2d 7374	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))))
25		mpteq1 5186	. . . . . . . . 9 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))
26	25	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄)))
27	26	oveq2d 7374	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑄))))
28	24, 27	eqeq12d 2751	. . . . . 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 3958	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
31	30	anbi2d 631	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐴)))
32		mpteq1 5186	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄)))
33	32	oveq2d 7374	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))))
34		mpteq1 5186	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ 𝑄) = (𝑘 ∈ 𝐴 ↦ 𝑄))
35	34	oveq2d 7374	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄)) = (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄)))
36	35	oveq2d 7374	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑃 · (𝑊 Σg (𝑘 ∈ 𝑎 ↦ 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))
37	33, 36	eqeq12d 2751	. . . . . 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 3933	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐺))
43		gsumvsca.g	. . . . . . . . . 10 ⊢ 𝐺 = (Scalar‘𝑊)
44		gsumvsca.t	. . . . . . . . . 10 ⊢ · = ( ·𝑠 ‘𝑊)
45		eqid 2735	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
46		gsumvsca.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑊)
47	43, 44, 45, 46	slmdvs0 33286	. . . . . . . . 9 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺)) → (𝑃 · 0 ) = 0 )
48	39, 42, 47	syl2anc 585	. . . . . . . 8 ⊢ (𝜑 → (𝑃 · 0 ) = 0 )
49	48	eqcomd 2741	. . . . . . 7 ⊢ (𝜑 → 0 = (𝑃 · 0 ))
50		mpt0 6633	. . . . . . . . 9 ⊢ (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄)) = ∅
51	50	oveq2i 7369	. . . . . . . 8 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑊 Σg ∅)
52	46	gsum0 18611	. . . . . . . 8 ⊢ (𝑊 Σg ∅) = 0
53	51, 52	eqtri 2758	. . . . . . 7 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = 0
54		mpt0 6633	. . . . . . . . . 10 ⊢ (𝑘 ∈ ∅ ↦ 𝑄) = ∅
55	54	oveq2i 7369	. . . . . . . . 9 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)) = (𝑊 Σg ∅)
56	55, 52	eqtri 2758	. . . . . . . 8 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄)) = 0
57	56	oveq2i 7369	. . . . . . 7 ⊢ (𝑃 · (𝑊 Σg (𝑘 ∈ ∅ ↦ 𝑄))) = (𝑃 · 0 )
58	49, 53, 57	3eqtr4g 2795	. . . . . 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 4129	. . . . . . . . 9 ⊢ 𝑒 ⊆ (𝑒 ∪ {𝑧})
61		sstr2 3939	. . . . . . . . 9 ⊢ (𝑒 ⊆ (𝑒 ∪ {𝑧}) → ((𝑒 ∪ {𝑧}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴))
62	60, 61	ax-mp 5	. . . . . . . 8 ⊢ ((𝑒 ∪ {𝑧}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴)
63	62	anim2i 618	. . . . . . 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 729	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑊 ∈ SLMod)
66	42	ad2antrl 729	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑃 ∈ (Base‘𝐺))
67		gsumvsca.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑊)
68		slmdcmn 33266	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
69	65, 68	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑊 ∈ CMnd)
70		vex 3443	. . . . . . . . . . . . 13 ⊢ 𝑒 ∈ V
71	70	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ∈ V)
72		simplrl 777	. . . . . . . . . . . . . 14 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝜑)
73		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑒 ∪ {𝑧}) ⊆ 𝐴)
74	73	unssad 4144	. . . . . . . . . . . . . . 15 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ⊆ 𝐴)
75	74	sselda 3932	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑄 ∈ 𝐵)
78	77	fmpttd 7060	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑘 ∈ 𝑒 ↦ 𝑄):𝑒⟶𝐵)
79		eqid 2735	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑒 ↦ 𝑄) = (𝑘 ∈ 𝑒 ↦ 𝑄)
80		simpll 767	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ∈ Fin)
81	46	fvexi 6847	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
82	81	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 0 ∈ V)
83	79, 80, 77, 82	fsuppmptdm 9281	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑘 ∈ 𝑒 ↦ 𝑄) finSupp 0 )
84	67, 46, 69, 71, 78, 83	gsumcl 19846	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ 𝑄)) ∈ 𝐵)
85	73	unssbd 4145	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → {𝑧} ⊆ 𝐴)
86		vex 3443	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
87	86	snss 4740	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
88	85, 87	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ 𝐴)
89	76	ralrimiva 3127	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑄 ∈ 𝐵)
90	89	ad2antrl 729	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ∀𝑘 ∈ 𝐴 𝑄 ∈ 𝐵)
91		rspcsbela 4389	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝑄 ∈ 𝐵) → ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵)
92	88, 90, 91	syl2anc 585	. . . . . . . . . . 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 33276	. . . . . . . . . . 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 1375	. . . . . . . . . 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 3872	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝑄
98	86	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ V)
99		simplr 769	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧 ∈ 𝑒)
100		csbeq1a 3862	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → 𝑄 = ⦋𝑧 / 𝑘⦌𝑄)
101	97, 67, 93, 69, 80, 77, 98, 99, 92, 100	gsumunsnf 19890	. . . . . . . . . . 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 7374	. . . . . . . . . 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 2897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝑃
105		nfcv 2897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 ·
106	104, 105, 97	nfov 7388	. . . . . . . . . . . 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 33275	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺) ∧ 𝑄 ∈ 𝐵) → (𝑃 · 𝑄) ∈ 𝐵)
110	107, 108, 77, 109	syl3anc 1374	. . . . . . . . . . . 12 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → (𝑃 · 𝑄) ∈ 𝐵)
111	67, 43, 44, 45	slmdvscl 33275	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺) ∧ ⦋𝑧 / 𝑘⦌𝑄 ∈ 𝐵) → (𝑃 · ⦋𝑧 / 𝑘⦌𝑄) ∈ 𝐵)
112	65, 66, 92, 111	syl3anc 1374	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑃 · ⦋𝑧 / 𝑘⦌𝑄) ∈ 𝐵)
113	100	oveq2d 7374	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑃 · 𝑄) = (𝑃 · ⦋𝑧 / 𝑘⦌𝑄))
114	106, 67, 93, 69, 80, 110, 98, 99, 112, 113	gsumunsnf 19890	. . . . . . . . . . 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 7373	. . . . . . . . . 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 2770	. . . . . . . . 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 2781	. . . . . . . 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 9092	. . . 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 705	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝜑) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))
126	1, 125	mpancom 689	1 ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = (𝑃 · (𝑊 Σg (𝑘 ∈ 𝐴 ↦ 𝑄))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3050  Vcvv 3439  ⦋csb 3848   ∪ cun 3898   ⊆ wss 3900  ∅c0 4284  {csn 4579   ↦ cmpt 5178  ‘cfv 6491  (class class class)co 7358  Fincfn 8885  Basecbs 17138  +_gcplusg 17179  Scalarcsca 17182   ·_𝑠 cvsca 17183  0_gc0g 17361   Σ_g cgsu 17362  CMndccmn 19711  SLModcslmd 33261

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-iin 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-se 5577  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-isom 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622  df-om 7809  df-1st 7933  df-2nd 7934  df-supp 8103  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-er 8635  df-en 8886  df-dom 8887  df-sdom 8888  df-fin 8889  df-fsupp 9267  df-oi 9417  df-card 9853  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11368  df-neg 11369  df-nn 12148  df-2 12210  df-n0 12404  df-z 12491  df-uz 12754  df-fz 13426  df-fzo 13573  df-seq 13927  df-hash 14256  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17139  df-ress 17160  df-plusg 17192  df-0g 17363  df-gsum 17364  df-mre 17507  df-mrc 17508  df-acs 17510  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-mulg 19000  df-cntz 19248  df-cmn 19713  df-srg 20124  df-slmd 33262

This theorem is referenced by: (None)