Theorem gsumzoppg 20014

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 19424	. . . . . . . 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 19426	. . . . . . . 8 ⊢ 0 = (0g‘𝑂)
8	7	gsumz 18895	. . . . . . 7 ⊢ ((𝑂 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
9	4, 5, 8	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
10	6	gsumz 18895	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
11	1, 5, 10	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
12	9, 11	eqtr4d 2807	. . . . 5 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
13	12	adantr 485	. . . 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 6896	. . . . . . 7 ⊢ 0 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
17		ssid 3967	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))
18	14, 5	fexd 7226	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
19		suppimacnv 8170	. . . . . . . . 9 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
20	18, 15, 19	sylancl 597	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
21	20	sseq1d 3976	. . . . . . 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 19978	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
24	23	oveq2d 7427	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
25	23	oveq2d 7427	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
26	13, 24, 25	3eqtr4d 2814	. . 3 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
27	26	ex 417	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
28		simprl 782	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
29		nnuz 12901	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
30	28, 29	eleqtrdi 2879	. . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵)
32		ffn 6706	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
33		dffn4 6799	. . . . . . . . . . . 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 6793	. . . . . . . . . . 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 485	. . . . . . . . . . . 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 18886	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
40		acsmre 17708	. . . . . . . . . . . 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 2769	. . . . . . . . . . 11 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
43	31	frnd 6715	. . . . . . . . . . 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 17681	. . . . . . . . . 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 6724	. . . . . . . . 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 6820	. . . . . . . . . . . 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 741	. . . . . . . . . . 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 6085	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
49	48, 31	fssdm 6726	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
50		f1ss 6782	. . . . . . . . . . 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 595	. . . . . . . . . 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 6775	. . . . . . . . . 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 6731	. . . . . . . . 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 595	. . . . . . . 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 17667	. . . . . . . . . 10 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹 ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
58	41, 43, 57	syl2anc 595	. . . . . . . . 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 19432	. . . . . . . . 9 ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂)
60	58, 59	eleqtrdi 2879	. . . . . . . 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 2769	. . . . . . . . . 10 ⊢ (+g‘𝑂) = (+g‘𝑂)
62	61	submcl 18870	. . . . . . . . 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 1136	. . . . . . . 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 591	. . . . . . 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 2769	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
66	65, 2, 61	oppgplus 19419	. . . . . . . . 9 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥)
67		gsumzoppg.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
68	67	adantr 485	. . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . 14 ⊢ (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
71	69, 42, 70	cntzspan 19914	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
72	37, 68, 71	syl2anc 595	. . . . . . . . . . . 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 19909	. . . . . . . . . . . . 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 3945	. . . . . . . . . 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 19399	. . . . . . . . . 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 591	. . . . . . . . 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 2823	. . . . . . . 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 471	. . . . . . 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 14087	. . . . . 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 19421	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
83		eqid 2769	. . . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉)
86	2, 69	oppgcntz 19434	. . . . . . . 8 ⊢ (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
87	68, 86	sseqtrdi 3985	. . . . . . 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 8173	. . . . . . . . . . 11 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
89	20, 88	eqsstrrdi 3990	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
90	14, 89	fssdmd 6725	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
91	90	adantr 485	. . . . . . . 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 595	. . . . . . 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 485	. . . . . . . . 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 6829	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })))
96		forn 6796	. . . . . . . . . . 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 3977	. . . . . . . . 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 741	. . . . . . . 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 2769	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
102	82, 7, 61, 83, 84, 85, 31, 87, 28, 92, 100, 101	gsumval3 19977	. . . . . 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 485	. . . . . . . 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 19977	. . . . . 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 2814	. . . . 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 461	. . . 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 1960	. . 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 458	. 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 9329	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
112	20, 111	eqeltrrd 2870	. . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
113		fz1f1o 15761	. . 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 873	1 ⊢ (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  {csn 4594   class class class wbr 5113   ↦ cmpt 5196  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   “ cima 5665   ∘ ccom 5666   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411   supp csupp 8156  Fincfn 8943   finSupp cfsupp 9321  1c1 11101  ℕcn 12233  ℤ_≥cuz 12862  ...cfz 13535  seqcseq 14037  ♯chash 14366  Basecbs 17269   ↾_s cress 17290  +_gcplusg 17310  0_gc0g 17492   Σ_g cgsu 17493  Moorecmre 17634  mrClscmrc 17635  ACScacs 17637  Mndcmnd 18792  SubMndcsubmnd 18840  Cntzccntz 19385  opp_gcoppg 19415  CMndccmn 19850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-tpos 8222  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-fzo 13683  df-seq 14038  df-hash 14367  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-0g 17494  df-gsum 17495  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-submnd 18842  df-cntz 19387  df-oppg 19416  df-cmn 19852

This theorem is referenced by:  gsumzinv  20015