Theorem gsumzoppg 19056

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 18474	. . . . . . . 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 18476	. . . . . . . 8 ⊢ 0 = (0g‘𝑂)
8	7	gsumz 17992	. . . . . . 7 ⊢ ((𝑂 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
9	4, 5, 8	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
10	6	gsumz 17992	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝐴 ∈ 𝑉) → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
11	1, 5, 10	syl2anc 586	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = 0 )
12	9, 11	eqtr4d 2857	. . . . 5 ⊢ (𝜑 → (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
13	12	adantr 483	. . . 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 6677	. . . . . . 7 ⊢ 0 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ (𝜑 → 0 ∈ V)
17		ssid 3987	. . . . . . 7 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))
18		fex 6981	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
19	14, 5, 18	syl2anc 586	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ V)
20		suppimacnv 7833	. . . . . . . . 9 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
21	19, 15, 20	sylancl 588	. . . . . . . 8 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
22	21	sseq1d 3996	. . . . . . 7 ⊢ (𝜑 → ((𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })) ↔ (◡𝐹 “ (V ∖ { 0 })) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
23	17, 22	mpbiri 260	. . . . . 6 ⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
24	14, 5, 16, 23	gsumcllem 19020	. . . . 5 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘 ∈ 𝐴 ↦ 0 ))
25	24	oveq2d 7164	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
26	24	oveq2d 7164	. . . 4 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 0 )))
27	13, 25, 26	3eqtr4d 2864	. . 3 ⊢ ((𝜑 ∧ (◡𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
28	27	ex 415	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
29		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
30		nnuz 12273	. . . . . . . 8 ⊢ ℕ = (ℤ≥‘1)
31	29, 30	eleqtrdi 2921	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ (ℤ≥‘1))
32	14	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶𝐵)
33		ffn 6507	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
34		dffn4 6589	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴–onto→ran 𝐹)
35	33, 34	sylib 220	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:𝐴–onto→ran 𝐹)
36		fof 6583	. . . . . . . . . . 11 ⊢ (𝐹:𝐴–onto→ran 𝐹 → 𝐹:𝐴⟶ran 𝐹)
37	32, 35, 36	3syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶ran 𝐹)
38	1	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
39		gsumzoppg.b	. . . . . . . . . . . . 13 ⊢ 𝐵 = (Base‘𝐺)
40	39	submacs 17983	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
41		acsmre 16915	. . . . . . . . . . . 12 ⊢ ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
42	38, 40, 41	3syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
43		eqid 2819	. . . . . . . . . . 11 ⊢ (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
44	32	frnd 6514	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ 𝐵)
45	42, 43, 44	mrcssidd 16888	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
46	37, 45	fssd 6521	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
47		f1of1 6607	. . . . . . . . . . . 12 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })))
48	47	ad2antll 727	. . . . . . . . . . 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 })))
49		cnvimass 5942	. . . . . . . . . . . 12 ⊢ (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
50	49, 32	fssdm 6523	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
51		f1ss 6573	. . . . . . . . . . 11 ⊢ ((𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→(◡𝐹 “ (V ∖ { 0 })) ∧ (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴)
52	48, 50, 51	syl2anc 586	. . . . . . . . . 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)
53		f1f 6568	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1→𝐴 → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))⟶𝐴)
54	52, 53	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)
55		fco 6524	. . . . . . . . 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))
56	46, 54, 55	syl2anc 586	. . . . . . . 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))
57	56	ffvelrnda 6844	. . . . . . 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))
58	43	mrccl 16874	. . . . . . . . . 10 ⊢ (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹 ⊆ 𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
59	42, 44, 58	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
60	2	oppgsubm 18482	. . . . . . . . 9 ⊢ (SubMnd‘𝐺) = (SubMnd‘𝑂)
61	59, 60	eleqtrdi 2921	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂))
62		eqid 2819	. . . . . . . . . 10 ⊢ (+g‘𝑂) = (+g‘𝑂)
63	62	submcl 17969	. . . . . . . . 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))
64	63	3expb 1115	. . . . . . . 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))
65	61, 64	sylan 582	. . . . . . 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))
66		gsumzoppg.c	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
67	66	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
68		gsumzoppg.z	. . . . . . . . . . . . . 14 ⊢ 𝑍 = (Cntz‘𝐺)
69		eqid 2819	. . . . . . . . . . . . . 14 ⊢ (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
70	68, 43, 69	cntzspan 18956	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺 ↾s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
71	38, 67, 70	syl2anc 586	. . . . . . . . . . . 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)
72	69, 68	submcmn2 18951	. . . . . . . . . . . . 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))))
73	59, 72	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))))
74	71, 73	mpbid 234	. . . . . . . . . . 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)))
75	74	sselda 3965	. . . . . . . . . 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)))
76		eqid 2819	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘𝐺)
77	76, 68	cntzi 18451	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))
78	75, 77	sylan 582	. . . . . . . . 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	76, 2, 62	oppgplus 18469	. . . . . . . . 9 ⊢ (𝑥(+g‘𝑂)𝑦) = (𝑦(+g‘𝐺)𝑥)
80	78, 79	syl6reqr 2873	. . . . . . . 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))
81	80	anasss 469	. . . . . . 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))
82	31, 57, 65, 81	seqfeq4 13411	. . . . . 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 })))))
83	2, 39	oppgbas 18471	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑂)
84		eqid 2819	. . . . . . 7 ⊢ (Cntz‘𝑂) = (Cntz‘𝑂)
85	38, 3	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝑂 ∈ Mnd)
86	5	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → 𝐴 ∈ 𝑉)
87	2, 68	oppgcntz 18484	. . . . . . . 8 ⊢ (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
88	67, 87	sseqtrdi 4015	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((Cntz‘𝑂)‘ran 𝐹))
89		suppssdm 7835	. . . . . . . . . . 11 ⊢ (𝐹 supp 0 ) ⊆ dom 𝐹
90	21, 89	eqsstrrdi 4020	. . . . . . . . . 10 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
91	14, 90	fssdmd 6522	. . . . . . . . 9 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
92	91	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (◡𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
93	48, 92, 51	syl2anc 586	. . . . . . 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)
94	22	adantr 483	. . . . . . . . 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 }))))
95	17, 94	mpbiri 260	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
96		f1ofo 6615	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })))
97		forn 6586	. . . . . . . . . . 11 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
98	96, 97	syl 17	. . . . . . . . . 10 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (◡𝐹 “ (V ∖ { 0 })))
99	98	sseq2d 3997	. . . . . . . . 9 ⊢ (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 }))))
100	99	ad2antll 727	. . . . . . . 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 }))))
101	95, 100	mpbird 259	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
102		eqid 2819	. . . . . . 7 ⊢ ((𝐹 ∘ 𝑓) supp 0 ) = ((𝐹 ∘ 𝑓) supp 0 )
103	83, 7, 62, 84, 85, 86, 32, 88, 29, 93, 101, 102	gsumval3 19019	. . . . . 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 })))))
104	23	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (◡𝐹 “ (V ∖ { 0 })))
105	104, 100	mpbird 259	. . . . . . 7 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
106	39, 6, 76, 68, 38, 86, 32, 67, 29, 93, 105, 102	gsumval3 19019	. . . . . 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 })))))
107	82, 103, 106	3eqtr4d 2864	. . . . 5 ⊢ ((𝜑 ∧ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
108	107	expr 459	. . . 4 ⊢ ((𝜑 ∧ (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
109	108	exlimdv 1928	. . 3 ⊢ ((𝜑 ∧ (♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
110	109	expimpd 456	. 2 ⊢ (𝜑 → (((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 }))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
111		gsumzoppg.n	. . . . 5 ⊢ (𝜑 → 𝐹 finSupp 0 )
112	111	fsuppimpd 8832	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin)
113	21, 112	eqeltrrd 2912	. . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ∈ Fin)
114		fz1f1o 15059	. . 3 ⊢ ((◡𝐹 “ (V ∖ { 0 })) ∈ Fin → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))))
115	113, 114	syl 17	. 2 ⊢ (𝜑 → ((◡𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(◡𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(◡𝐹 “ (V ∖ { 0 }))))–1-1-onto→(◡𝐹 “ (V ∖ { 0 })))))
116	28, 110, 115	mpjaod 856	1 ⊢ (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1531  ∃wex 1774   ∈ wcel 2108  Vcvv 3493   ∖ cdif 3931   ⊆ wss 3934  ∅c0 4289  {csn 4559   class class class wbr 5057   ↦ cmpt 5137  ^◡ccnv 5547  dom cdm 5548  ran crn 5549   “ cima 5551   ∘ ccom 5552   Fn wfn 6343  ⟶wf 6344  –1-1→wf1 6345  –onto→wfo 6346  –1-1-onto→wf1o 6347  ‘cfv 6348  (class class class)co 7148   supp csupp 7822  Fincfn 8501   finSupp cfsupp 8825  1c1 10530  ℕcn 11630  ℤ_≥cuz 12235  ...cfz 12884  seqcseq 13361  ♯chash 13682  Basecbs 16475   ↾_s cress 16476  +_gcplusg 16557  0_gc0g 16705   Σ_g cgsu 16706  Moorecmre 16845  mrClscmrc 16846  ACScacs 16848  Mndcmnd 17903  SubMndcsubmnd 17947  Cntzccntz 18437  opp_gcoppg 18465  CMndccmn 18898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-supp 7823  df-tpos 7884  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-1o 8094  df-oadd 8098  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-fsupp 8826  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-fzo 13026  df-seq 13362  df-hash 13683  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-0g 16707  df-gsum 16708  df-mre 16849  df-mrc 16850  df-acs 16852  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-cntz 18439  df-oppg 18466  df-cmn 18900

This theorem is referenced by:  gsumzinv  19057