Theorem gsumvsca2 29114

Description: Scalar product of a finite group sum for a left module over a semiring. (Contributed by Thierry Arnoux, 16-Mar-2018.) (Proof shortened by AV, 12-Dec-2019.)

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)
gsumvsca2.n	⊢ (𝜑 → 𝑄 ∈ 𝐵)
gsumvsca2.c	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐾)

Assertion

Ref	Expression
gsumvsca2	⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄))

Distinct variable groups:   · ,𝑘   𝐴,𝑘   𝑘,𝑊   𝜑,𝑘   𝐵,𝑘   𝑘,𝐺   𝑘,𝐾   𝑄,𝑘

Allowed substitution hints:   𝑃(𝑘)   + (𝑘)   0 (𝑘)

Proof of Theorem gsumvsca2

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 3587	. . 3 ⊢ 𝐴 ⊆ 𝐴
3		sseq1 3589	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
4	3	anbi2d 736	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ ∅ ⊆ 𝐴)))
5		mpteq1 4665	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄)))
6	5	oveq2d 6565	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))))
7		mpteq1 4665	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑘 ∈ 𝑎 ↦ 𝑃) = (𝑘 ∈ ∅ ↦ 𝑃))
8	7	oveq2d 6565	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) = (𝐺 Σg (𝑘 ∈ ∅ ↦ 𝑃)))
9	8	oveq1d 6564	. . . . . . 7 ⊢ (𝑎 = ∅ → ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄) = ((𝐺 Σg (𝑘 ∈ ∅ ↦ 𝑃)) · 𝑄))
10	6, 9	eqeq12d 2625	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄) ↔ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ ∅ ↦ 𝑃)) · 𝑄)))
11	4, 10	imbi12d 333	. . . . 5 ⊢ (𝑎 = ∅ → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄)) ↔ ((𝜑 ∧ ∅ ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ ∅ ↦ 𝑃)) · 𝑄))))
12		sseq1 3589	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑎 ⊆ 𝐴 ↔ 𝑒 ⊆ 𝐴))
13	12	anbi2d 736	. . . . . 6 ⊢ (𝑎 = 𝑒 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝑒 ⊆ 𝐴)))
14		mpteq1 4665	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄)))
15	14	oveq2d 6565	. . . . . . 7 ⊢ (𝑎 = 𝑒 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))))
16		mpteq1 4665	. . . . . . . . 9 ⊢ (𝑎 = 𝑒 → (𝑘 ∈ 𝑎 ↦ 𝑃) = (𝑘 ∈ 𝑒 ↦ 𝑃))
17	16	oveq2d 6565	. . . . . . . 8 ⊢ (𝑎 = 𝑒 → (𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) = (𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)))
18	17	oveq1d 6564	. . . . . . 7 ⊢ (𝑎 = 𝑒 → ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄))
19	15, 18	eqeq12d 2625	. . . . . 6 ⊢ (𝑎 = 𝑒 → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄) ↔ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)))
20	13, 19	imbi12d 333	. . . . 5 ⊢ (𝑎 = 𝑒 → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄)) ↔ ((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄))))
21		sseq1 3589	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑎 ⊆ 𝐴 ↔ (𝑒 ∪ {𝑧}) ⊆ 𝐴))
22	21	anbi2d 736	. . . . . 6 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)))
23		mpteq1 4665	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄)))
24	23	oveq2d 6565	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))))
25		mpteq1 4665	. . . . . . . . 9 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝑘 ∈ 𝑎 ↦ 𝑃) = (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃))
26	25	oveq2d 6565	. . . . . . . 8 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) = (𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)))
27	26	oveq1d 6564	. . . . . . 7 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄) = ((𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) · 𝑄))
28	24, 27	eqeq12d 2625	. . . . . 6 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄) ↔ (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) · 𝑄)))
29	22, 28	imbi12d 333	. . . . 5 ⊢ (𝑎 = (𝑒 ∪ {𝑧}) → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄)) ↔ ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) · 𝑄))))
30		sseq1 3589	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
31	30	anbi2d 736	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝜑 ∧ 𝑎 ⊆ 𝐴) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐴)))
32		mpteq1 4665	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄)) = (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄)))
33	32	oveq2d 6565	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))))
34		mpteq1 4665	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑘 ∈ 𝑎 ↦ 𝑃) = (𝑘 ∈ 𝐴 ↦ 𝑃))
35	34	oveq2d 6565	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)))
36	35	oveq1d 6564	. . . . . . 7 ⊢ (𝑎 = 𝐴 → ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄))
37	33, 36	eqeq12d 2625	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄) ↔ (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄)))
38	31, 37	imbi12d 333	. . . . 5 ⊢ (𝑎 = 𝐴 → (((𝜑 ∧ 𝑎 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑎 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑎 ↦ 𝑃)) · 𝑄)) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄))))
39		gsumvsca.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ SLMod)
40		gsumvsca2.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
41		gsumvsca.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝑊)
42		gsumvsca.g	. . . . . . . . . 10 ⊢ 𝐺 = (Scalar‘𝑊)
43		gsumvsca.t	. . . . . . . . . 10 ⊢ · = ( ·𝑠 ‘𝑊)
44		eqid 2610	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
45		gsumvsca.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑊)
46	41, 42, 43, 44, 45	slmd0vs 29108	. . . . . . . . 9 ⊢ ((𝑊 ∈ SLMod ∧ 𝑄 ∈ 𝐵) → ((0g‘𝐺) · 𝑄) = 0 )
47	39, 40, 46	syl2anc 691	. . . . . . . 8 ⊢ (𝜑 → ((0g‘𝐺) · 𝑄) = 0 )
48	47	eqcomd 2616	. . . . . . 7 ⊢ (𝜑 → 0 = ((0g‘𝐺) · 𝑄))
49		mpt0 5934	. . . . . . . . 9 ⊢ (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄)) = ∅
50	49	oveq2i 6560	. . . . . . . 8 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = (𝑊 Σg ∅)
51	45	gsum0 17101	. . . . . . . 8 ⊢ (𝑊 Σg ∅) = 0
52	50, 51	eqtri 2632	. . . . . . 7 ⊢ (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = 0
53		mpt0 5934	. . . . . . . . . 10 ⊢ (𝑘 ∈ ∅ ↦ 𝑃) = ∅
54	53	oveq2i 6560	. . . . . . . . 9 ⊢ (𝐺 Σg (𝑘 ∈ ∅ ↦ 𝑃)) = (𝐺 Σg ∅)
55	44	gsum0 17101	. . . . . . . . 9 ⊢ (𝐺 Σg ∅) = (0g‘𝐺)
56	54, 55	eqtri 2632	. . . . . . . 8 ⊢ (𝐺 Σg (𝑘 ∈ ∅ ↦ 𝑃)) = (0g‘𝐺)
57	56	oveq1i 6559	. . . . . . 7 ⊢ ((𝐺 Σg (𝑘 ∈ ∅ ↦ 𝑃)) · 𝑄) = ((0g‘𝐺) · 𝑄)
58	48, 52, 57	3eqtr4g 2669	. . . . . 6 ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ ∅ ↦ 𝑃)) · 𝑄))
59	58	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∅ ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ ∅ ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ ∅ ↦ 𝑃)) · 𝑄))
60		ssun1 3738	. . . . . . . . 9 ⊢ 𝑒 ⊆ (𝑒 ∪ {𝑧})
61		sstr2 3575	. . . . . . . . 9 ⊢ (𝑒 ⊆ (𝑒 ∪ {𝑧}) → ((𝑒 ∪ {𝑧}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴))
62	60, 61	ax-mp 5	. . . . . . . 8 ⊢ ((𝑒 ∪ {𝑧}) ⊆ 𝐴 → 𝑒 ⊆ 𝐴)
63	62	anim2i 591	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝜑 ∧ 𝑒 ⊆ 𝐴))
64	63	imim1i 61	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)))
65	39	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑊 ∈ SLMod)
66		eqid 2610	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
67	42	slmdsrg 29091	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ SLMod → 𝐺 ∈ SRing)
68		srgcmn 18331	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ SRing → 𝐺 ∈ CMnd)
69	65, 67, 68	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝐺 ∈ CMnd)
70		vex 3176	. . . . . . . . . . . . 13 ⊢ 𝑒 ∈ V
71	70	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ∈ V)
72		simplrl 796	. . . . . . . . . . . . . 14 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝜑)
73		simprr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑒 ∪ {𝑧}) ⊆ 𝐴)
74	73	unssad 3752	. . . . . . . . . . . . . . 15 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ⊆ 𝐴)
75	74	sselda 3568	. . . . . . . . . . . . . 14 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑘 ∈ 𝐴)
76		gsumvsca.k	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ⊆ (Base‘𝐺))
77	76	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐾 ⊆ (Base‘𝐺))
78		gsumvsca2.c	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ 𝐾)
79	77, 78	sseldd 3569	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑃 ∈ (Base‘𝐺))
80	72, 75, 79	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑃 ∈ (Base‘𝐺))
81		eqid 2610	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ 𝑒 ↦ 𝑃) = (𝑘 ∈ 𝑒 ↦ 𝑃)
82	80, 81	fmptd 6292	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑘 ∈ 𝑒 ↦ 𝑃):𝑒⟶(Base‘𝐺))
83		simpll 786	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑒 ∈ Fin)
84	72, 75, 78	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑃 ∈ 𝐾)
85		fvex 6113	. . . . . . . . . . . . . 14 ⊢ (0g‘𝐺) ∈ V
86	85	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (0g‘𝐺) ∈ V)
87	81, 83, 84, 86	fsuppmptdm 8169	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑘 ∈ 𝑒 ↦ 𝑃) finSupp (0g‘𝐺))
88	66, 44, 69, 71, 82, 87	gsumcl 18139	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) ∈ (Base‘𝐺))
89	73	unssbd 3753	. . . . . . . . . . . . 13 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → {𝑧} ⊆ 𝐴)
90		vex 3176	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
91	90	snss 4259	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 ↔ {𝑧} ⊆ 𝐴)
92	89, 91	sylibr 223	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ 𝐴)
93	79	ralrimiva 2949	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝑃 ∈ (Base‘𝐺))
94	93	ad2antrl 760	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ∀𝑘 ∈ 𝐴 𝑃 ∈ (Base‘𝐺))
95		rspcsbela 3958	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑘 ∈ 𝐴 𝑃 ∈ (Base‘𝐺)) → ⦋𝑧 / 𝑘⦌𝑃 ∈ (Base‘𝐺))
96	92, 94, 95	syl2anc 691	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ⦋𝑧 / 𝑘⦌𝑃 ∈ (Base‘𝐺))
97	40	ad2antrl 760	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑄 ∈ 𝐵)
98		gsumvsca.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑊)
99		eqid 2610	. . . . . . . . . . . 12 ⊢ (+g‘𝐺) = (+g‘𝐺)
100	41, 98, 42, 43, 66, 99	slmdvsdir 29100	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ SLMod ∧ ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) ∈ (Base‘𝐺) ∧ ⦋𝑧 / 𝑘⦌𝑃 ∈ (Base‘𝐺) ∧ 𝑄 ∈ 𝐵)) → (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃))(+g‘𝐺)⦋𝑧 / 𝑘⦌𝑃) · 𝑄) = (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄) + (⦋𝑧 / 𝑘⦌𝑃 · 𝑄)))
101	65, 88, 96, 97, 100	syl13anc 1320	. . . . . . . . . 10 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃))(+g‘𝐺)⦋𝑧 / 𝑘⦌𝑃) · 𝑄) = (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄) + (⦋𝑧 / 𝑘⦌𝑃 · 𝑄)))
102	101	adantr 480	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃))(+g‘𝐺)⦋𝑧 / 𝑘⦌𝑃) · 𝑄) = (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄) + (⦋𝑧 / 𝑘⦌𝑃 · 𝑄)))
103		nfcsb1v 3515	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋𝑧 / 𝑘⦌𝑃
104	90	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑧 ∈ V)
105		simplr 788	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ¬ 𝑧 ∈ 𝑒)
106		csbeq1a 3508	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → 𝑃 = ⦋𝑧 / 𝑘⦌𝑃)
107	103, 66, 99, 69, 83, 80, 104, 105, 96, 106	gsumunsnf 18181	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃))(+g‘𝐺)⦋𝑧 / 𝑘⦌𝑃))
108	107	oveq1d 6564	. . . . . . . . . 10 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → ((𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) · 𝑄) = (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃))(+g‘𝐺)⦋𝑧 / 𝑘⦌𝑃) · 𝑄))
109	108	adantr 480	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → ((𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) · 𝑄) = (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃))(+g‘𝐺)⦋𝑧 / 𝑘⦌𝑃) · 𝑄))
110		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 ·
111		nfcv 2751	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝑄
112	103, 110, 111	nfov 6575	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(⦋𝑧 / 𝑘⦌𝑃 · 𝑄)
113		slmdcmn 29089	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd)
114	65, 113	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → 𝑊 ∈ CMnd)
115	72, 39	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑊 ∈ SLMod)
116	72, 40	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → 𝑄 ∈ 𝐵)
117	41, 42, 43, 66	slmdvscl 29098	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ SLMod ∧ 𝑃 ∈ (Base‘𝐺) ∧ 𝑄 ∈ 𝐵) → (𝑃 · 𝑄) ∈ 𝐵)
118	115, 80, 116, 117	syl3anc 1318	. . . . . . . . . . . 12 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ 𝑘 ∈ 𝑒) → (𝑃 · 𝑄) ∈ 𝐵)
119	41, 42, 43, 66	slmdvscl 29098	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ SLMod ∧ ⦋𝑧 / 𝑘⦌𝑃 ∈ (Base‘𝐺) ∧ 𝑄 ∈ 𝐵) → (⦋𝑧 / 𝑘⦌𝑃 · 𝑄) ∈ 𝐵)
120	65, 96, 97, 119	syl3anc 1318	. . . . . . . . . . . 12 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (⦋𝑧 / 𝑘⦌𝑃 · 𝑄) ∈ 𝐵)
121	106	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑧 → (𝑃 · 𝑄) = (⦋𝑧 / 𝑘⦌𝑃 · 𝑄))
122	112, 41, 98, 114, 83, 118, 104, 105, 120, 121	gsumunsnf 18181	. . . . . . . . . . 11 ⊢ (((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) + (⦋𝑧 / 𝑘⦌𝑃 · 𝑄)))
123	122	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) + (⦋𝑧 / 𝑘⦌𝑃 · 𝑄)))
124		simpr 476	. . . . . . . . . . 11 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄))
125	124	oveq1d 6564	. . . . . . . . . 10 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) + (⦋𝑧 / 𝑘⦌𝑃 · 𝑄)) = (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄) + (⦋𝑧 / 𝑘⦌𝑃 · 𝑄)))
126	123, 125	eqtrd 2644	. . . . . . . . 9 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = (((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄) + (⦋𝑧 / 𝑘⦌𝑃 · 𝑄)))
127	102, 109, 126	3eqtr4rd 2655	. . . . . . . 8 ⊢ ((((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) ∧ (𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴)) ∧ (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) · 𝑄))
128	127	exp31 628	. . . . . . 7 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → ((𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) · 𝑄))))
129	128	a2d 29	. . . . . 6 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) → (((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) · 𝑄))))
130	64, 129	syl5 33	. . . . 5 ⊢ ((𝑒 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑒) → (((𝜑 ∧ 𝑒 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝑒 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝑒 ↦ 𝑃)) · 𝑄)) → ((𝜑 ∧ (𝑒 ∪ {𝑧}) ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ (𝑒 ∪ {𝑧}) ↦ 𝑃)) · 𝑄))))
131	11, 20, 29, 38, 59, 130	findcard2s 8086	. . . 4 ⊢ (𝐴 ∈ Fin → ((𝜑 ∧ 𝐴 ⊆ 𝐴) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄)))
132	131	imp 444	. . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝜑 ∧ 𝐴 ⊆ 𝐴)) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄))
133	2, 132	mpanr2 716	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝜑) → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄))
134	1, 133	mpancom 700	1 ⊢ (𝜑 → (𝑊 Σg (𝑘 ∈ 𝐴 ↦ (𝑃 · 𝑄))) = ((𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑃)) · 𝑄))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⦋csb 3499   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  Basecbs 15695  +_gcplusg 15768  Scalarcsca 15771   ·_𝑠 cvsca 15772  0_gc0g 15923   Σ_g cgsu 15924  CMndccmn 18016  SRingcsrg 18328  SLModcslmd 29084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-srg 18329  df-slmd 29085

This theorem is referenced by: (None)