Theorem gsumzoppg 19962

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 19373	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → 𝑂 ∈ Mnd)
4	1, 3	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑂 ∈ Mnd)
5		gsumzoppg.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		gsumzoppg.0	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
7	2, 6	oppgid 19375	. . . . . . . 8 ⊢ 0 = (0g‘𝑂)
8	7	gsumz 18849	. . . . . . 7 ⊢ ((𝑂 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
9	4, 5, 8	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
10	6	gsumz 18849	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
11	1, 5, 10	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
12	9, 11	eqtr4d 2780	. . . . 5 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
13	12	adantr 480	. . . 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 6920	. . . . . . 7 ⊢ 0 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
17		ssid 4006	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))
18	14, 5	fexd 7247	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
19		suppimacnv 8199	. . . . . . . . 9 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
20	18, 15, 19	sylancl 586	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
21	20	sseq1d 4015	. . . . . . 7 ⊢ (𝜑 → ((𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })) ↔ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
22	17, 21	mpbiri 258	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
23	14, 5, 16, 22	gsumcllem 19926	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
24	23	oveq2d 7447	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
25	23	oveq2d 7447	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
26	13, 24, 25	3eqtr4d 2787	. . 3 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
27	26	ex 412	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
28		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
29		nnuz 12921	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
30	28, 29	eleqtrdi 2851	. . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵)
32		ffn 6736	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
33		dffn4 6826	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
34	32, 33	sylib 218	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹)
35		fof 6820	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–onto→ran 𝐹 → 𝐹:𝐴⟶ran 𝐹)
36	31, 34, 35	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶ran 𝐹)
37	1	adantr 480	. . . . . . . . . . . 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 18840	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
40		acsmre 17695	. . . . . . . . . . . 12 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
41	37, 39, 40	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
42		eqid 2737	. . . . . . . . . . 11 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
43	31	frnd 6744	. . . . . . . . . . 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 17668	. . . . . . . . . 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 6753	. . . . . . . . 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 6847	. . . . . . . . . . . 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 729	. . . . . . . . . . 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 6100	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
49	48, 31	fssdm 6755	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
50		f1ss 6809	. . . . . . . . . . 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 584	. . . . . . . . . 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 6804	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
53	51, 52	syl 17	. . . . . . . . 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 6760	. . . . . . . . 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 584	. . . . . . . 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 7104	. . . . . . 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 17654	. . . . . . . . . 10 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹 ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
58	41, 43, 57	syl2anc 584	. . . . . . . . 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 19381	. . . . . . . . 9 ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂)
60	58, 59	eleqtrdi 2851	. . . . . . . 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 2737	. . . . . . . . . 10 ⊢ (+g‘𝑂) = (+g‘𝑂)
62	61	submcl 18825	. . . . . . . . 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 1121	. . . . . . . 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 580	. . . . . . 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 2737	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
66	65, 2, 61	oppgplus 19367	. . . . . . . . 9 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥)
67		gsumzoppg.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
68	67	adantr 480	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . 14 ⊢ (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
71	69, 42, 70	cntzspan 19862	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
72	37, 68, 71	syl2anc 584	. . . . . . . . . . . 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 19857	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 232	. . . . . . . . . . 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 3983	. . . . . . . . . 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 19347	. . . . . . . . . 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 580	. . . . . . . . 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 2796	. . . . . . . 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 466	. . . . . . 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 14092	. . . . . 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 19370	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
83		eqid 2737	. . . . . . 7 ⊢ (Cntz‘𝑂) = (Cntz‘𝑂)
84	37, 3	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑂 ∈ Mnd)
85	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉)
86	2, 69	oppgcntz 19383	. . . . . . . 8 ⊢ (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
87	68, 86	sseqtrdi 4024	. . . . . . 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 8202	. . . . . . . . . . 11 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
89	20, 88	eqsstrrdi 4029	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
90	14, 89	fssdmd 6754	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
91	90	adantr 480	. . . . . . . 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 584	. . . . . . 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 480	. . . . . . . . 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 258	. . . . . . . 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 6855	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })))
96		forn 6823	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
97	95, 96	syl 17	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
98	97	sseq2d 4016	. . . . . . . . 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 729	. . . . . . . 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 257	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
101		eqid 2737	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
102	82, 7, 61, 83, 84, 85, 31, 87, 28, 92, 100, 101	gsumval3 19925	. . . . . 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 480	. . . . . . . 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 257	. . . . . . 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 19925	. . . . . 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 2787	. . . . 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 456	. . . 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 1933	. . 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 453	. 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 9409	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
112	20, 111	eqeltrrd 2842	. . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
113		fz1f1o 15746	. . 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 17	. 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 861	1 ⊢ (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  {csn 4626   class class class wbr 5143   ↦ cmpt 5225  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688   ∘ ccom 5689   Fn wfn 6556  ⟶wf 6557  –1-1→wf1 6558  –onto→wfo 6559  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   supp csupp 8185  Fincfn 8985   finSupp cfsupp 9401  1c1 11156  ℕcn 12266  ℤ_≥cuz 12878  ...cfz 13547  seqcseq 14042  ♯chash 14369  Basecbs 17247   ↾_s cress 17274  +_gcplusg 17297  0_gc0g 17484   Σ_g cgsu 17485  Moorecmre 17625  mrClscmrc 17626  ACScacs 17628  Mndcmnd 18747  SubMndcsubmnd 18795  Cntzccntz 19333  opp_gcoppg 19363  CMndccmn 19798

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-cntz 19335  df-oppg 19364  df-cmn 19800

This theorem is referenced by:  gsumzinv  19963