Theorem gsumzoppg 20097

Description: The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 6-Jun-2019.)

Hypotheses

Ref	Expression
gsumzoppg.b	⊢ 𝐵 = (Base‘𝐺)
gsumzoppg.0	⊢ 0 = (0g‘𝐺)
gsumzoppg.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumzoppg.o	⊢ 𝑂 = (oppg‘𝐺)
gsumzoppg.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumzoppg.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumzoppg.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumzoppg.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzoppg.n	⊢ (𝜑 → 𝐹 finSupp 0 )

Assertion

Ref	Expression
gsumzoppg	⊢ (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))

Proof of Theorem gsumzoppg

Dummy variables 𝑓 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumzoppg.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		gsumzoppg.o	. . . . . . . . 9 ⊢ 𝑂 = (oppg‘𝐺)
3	2	oppgmnd 19507	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → 𝑂 ∈ Mnd)
4	1, 3	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Mnd)
5		gsumzoppg.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		gsumzoppg.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
7	2, 6	oppgid 19509	. . . . . . . 8 ⊢ 0 = (0g‘𝑂)
8	7	gsumz 18971	. . . . . . 7 ⊢ ((𝑂 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
9	4, 5, 8	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
10	6	gsumz 18971	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
11	1, 5, 10	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
12	9, 11	eqtr4d 2798	. . . . 5 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
13	12	adantr 486	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
14		gsumzoppg.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
15	6	fvexi 6895	. . . . . . 7 ⊢ 0 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
17		ssid 3953	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))
18	14, 5	fexd 7229	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
19		suppimacnv 8177	. . . . . . . . 9 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
20	18, 15, 19	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
21	20	sseq1d 3962	. . . . . . 7 ⊢ (𝜑 → ((𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })) ↔ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
22	17, 21	mpbiri 261	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
23	14, 5, 16, 22	gsumcllem 20061	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
24	23	oveq2d 7432	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
25	23	oveq2d 7432	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
26	13, 24, 25	3eqtr4d 2805	. . 3 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
27	26	ex 418	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
28		simprl 783	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
29		nnuz 12951	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
30	28, 29	eleqtrdi 2870	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ (ℤ≥‘1))
31	14	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵)
32		ffn 6705	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
33		dffn4 6798	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
34	32, 33	sylib 221	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹)
35		fof 6792	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–onto→ran 𝐹 → 𝐹:𝐴⟶ran 𝐹)
36	31, 34, 35	3syl 19	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶ran 𝐹)
37	1	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
38		gsumzoppg.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
39	38	submacs 18962	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
40		acsmre 17765	. . . . . . . . . . . 12 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
41	37, 39, 40	3syl 19	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
42		eqid 2760	. . . . . . . . . . 11 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
43	31	frnd 6714	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ 𝐵)
44	41, 42, 43	mrcssidd 17738	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
45	36, 44	fssd 6723	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
46		f1of1 6819	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })))
47	46	ad2antll 742	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })))
48		cnvimass 6080	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
49	48, 31	fssdm 6725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
50		f1ss 6781	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })) ∧ (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
51	47, 49, 50	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
52		f1f 6774	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
53	51, 52	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
54		fco 6730	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹 ∘ 𝑓):(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
55	45, 53, 54	syl2anc 596	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 ∘ 𝑓):(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
56	55	ffvelcdmda 7080	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))) → ((𝐹 ∘ 𝑓)‘𝑥) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
57	42	mrccl 17724	. . . . . . . . . 10 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹 ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
58	41, 43, 57	syl2anc 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
59	2	oppgsubm 19515	. . . . . . . . 9 ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂)
60	58, 59	eleqtrdi 2870	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂))
61		eqid 2760	. . . . . . . . . 10 ⊢ (+g‘𝑂) = (+g‘𝑂)
62	61	submcl 18946	. . . . . . . . 9 ⊢ ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂) ∧ 𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g‘𝑂)𝑦) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
63	62	3expb 1138	. . . . . . . 8 ⊢ ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂) ∧ (𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))) → (𝑥(+g‘𝑂)𝑦) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
64	60, 63	sylan 592	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ (𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))) → (𝑥(+g‘𝑂)𝑦) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
65		eqid 2760	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
66	65, 2, 61	oppgplus 19502	. . . . . . . . 9 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥)
67		gsumzoppg.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
68	67	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
69		gsumzoppg.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (Cntz‘𝐺)
70		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
71	69, 42, 70	cntzspan 19997	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
72	37, 68, 71	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
73	70, 69	submcmn2 19992	. . . . . . . . . . . . 13 ⊢ (((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → ((𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
74	58, 73	syl 18	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
75	72, 74	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))
76	75	sselda 3931	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → 𝑥 ∈ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)))
77	65, 69	cntzi 19482	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
78	76, 77	sylan 592	. . . . . . . . 9 ⊢ ((((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
79	66, 78	eqtr4id 2814	. . . . . . . 8 ⊢ ((((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g‘𝑂)𝑦) = (𝑥(+g‘𝐺)𝑦))
80	79	anasss 472	. . . . . . 7 ⊢ (((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) ∧ (𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))) → (𝑥(+g‘𝑂)𝑦) = (𝑥(+g‘𝐺)𝑦))
81	30, 56, 64, 80	seqfeq4 14140	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (seq1((+g‘𝑂), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))
82	2, 38	oppgbas 19504	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
83		eqid 2760	. . . . . . 7 ⊢ (Cntz‘𝑂) = (Cntz‘𝑂)
84	37, 3	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑂 ∈ Mnd)
85	5	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉)
86	2, 69	oppgcntz 19517	. . . . . . . 8 ⊢ (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
87	68, 86	sseqtrdi 3971	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((Cntz‘𝑂)‘ran 𝐹))
88		suppssdm 8180	. . . . . . . . . . 11 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
89	20, 88	eqsstrrdi 3976	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
90	14, 89	fssdmd 6724	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
91	90	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
92	47, 91, 50	syl2anc 596	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
93	21	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })) ↔ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
94	17, 93	mpbiri 261	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
95		f1ofo 6828	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })))
96		forn 6795	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
97	95, 96	syl 18	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
98	97	sseq2d 3963	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
99	98	ad2antll 742	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
100	94, 99	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
101		eqid 2760	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
102	82, 7, 61, 83, 84, 85, 31, 87, 28, 92, 100, 101	gsumval3 20060	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (seq1((+g‘𝑂), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))
103	22	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
104	103, 99	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
105	38, 6, 65, 69, 37, 85, 31, 68, 28, 92, 104, 101	gsumval3 20060	. . . . . 6 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐺 Σg 𝐹) = (seq1((+g‘𝐺), (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ { 0 })))))
106	81, 102, 105	3eqtr4d 2805	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
107	106	expr 462	. . . 4 ⊢ ((𝜑 ∧ (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
108	107	exlimdv 1966	. . 3 ⊢ ((𝜑 ∧ (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
109	108	expimpd 459	. 2 ⊢ (𝜑 → (((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
110		gsumzoppg.n	. . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 )
111	110	fsuppimpd 9346	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
112	20, 111	eqeltrrd 2861	. . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
113		fz1f1o 15821	. . 3 ⊢ ((◡𝐹 “ (V ∖ { 0 })) ∈ Fin → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))))
114	112, 113	syl 18	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))))
115	27, 109, 114	mpjaod 874	1 ⊢ (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3450   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  {csn 4584   class class class wbr 5103   ↦ cmpt 5186  ^◡ccnv 5654  dom cdm 5655  ran crn 5656   “ cima 5658   ∘ ccom 5659   Fn wfn 6530  ⟶wf 6531  –1-1→wf1 6532  –onto→wfo 6533  –1-1-onto→wf1o 6534  ‘cfv 6535  (class class class)co 7416   supp csupp 8163  Fincfn 8959   finSupp cfsupp 9338  1c1 11150  ℕcn 12282  ℤ_≥cuz 12912  ...cfz 13586  seqcseq 14090  ♯chash 14419  Basecbs 17326   ↾_s cress 17347  +_gcplusg 17367  0_gc0g 17549   Σ_g cgsu 17550  Moorecmre 17691  mrClscmrc 17692  ACScacs 17694  Mndcmnd 18862  SubMndcsubmnd 18916  Cntzccntz 19468  opp_gcoppg 19498  CMndccmn 19933

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 7742  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226

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-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 6301  df-ord 6362  df-on 6363  df-lim 6364  df-suc 6365  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7992  df-2nd 7993  df-supp 8164  df-tpos 8229  df-frecs 8285  df-wrecs 8316  df-recs 8365  df-rdg 8404  df-1o 8462  df-2o 8463  df-er 8703  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-fsupp 9339  df-oi 9489  df-card 9969  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11492  df-neg 11493  df-nn 12283  df-2 12352  df-n0 12554  df-z 12641  df-uz 12913  df-fz 13587  df-fzo 13735  df-seq 14091  df-hash 14420  df-sets 17281  df-slot 17299  df-ndx 17311  df-base 17327  df-ress 17348  df-plusg 17380  df-0g 17551  df-gsum 17552  df-mre 17695  df-mrc 17696  df-acs 17698  df-mgm 18755  df-sgrp 18847  df-mnd 18863  df-submnd 18918  df-cntz 19470  df-oppg 19499  df-cmn 19935

This theorem is used by:  gsumzinv  20098