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Theorem gsumzoppg 19721
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 = (0g𝐺)
gsumzoppg.z 𝑍 = (Cntz‘𝐺)
gsumzoppg.o 𝑂 = (oppg𝐺)
gsumzoppg.g (𝜑𝐺 ∈ Mnd)
gsumzoppg.a (𝜑𝐴𝑉)
gsumzoppg.f (𝜑𝐹:𝐴𝐵)
gsumzoppg.c (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumzoppg.n (𝜑𝐹 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.
StepHypRef Expression
1 gsumzoppg.g . . . . . . . 8 (𝜑𝐺 ∈ Mnd)
2 gsumzoppg.o . . . . . . . . 9 𝑂 = (oppg𝐺)
32oppgmnd 19135 . . . . . . . 8 (𝐺 ∈ Mnd → 𝑂 ∈ Mnd)
41, 3syl 17 . . . . . . 7 (𝜑𝑂 ∈ Mnd)
5 gsumzoppg.a . . . . . . 7 (𝜑𝐴𝑉)
6 gsumzoppg.0 . . . . . . . . 9 0 = (0g𝐺)
72, 6oppgid 19137 . . . . . . . 8 0 = (0g𝑂)
87gsumz 18646 . . . . . . 7 ((𝑂 ∈ Mnd ∧ 𝐴𝑉) → (𝑂 Σg (𝑘𝐴0 )) = 0 )
94, 5, 8syl2anc 584 . . . . . 6 (𝜑 → (𝑂 Σg (𝑘𝐴0 )) = 0 )
106gsumz 18646 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
111, 5, 10syl2anc 584 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
129, 11eqtr4d 2779 . . . . 5 (𝜑 → (𝑂 Σg (𝑘𝐴0 )) = (𝐺 Σg (𝑘𝐴0 )))
1312adantr 481 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg (𝑘𝐴0 )) = (𝐺 Σg (𝑘𝐴0 )))
14 gsumzoppg.f . . . . . 6 (𝜑𝐹:𝐴𝐵)
156fvexi 6856 . . . . . . 7 0 ∈ V
1615a1i 11 . . . . . 6 (𝜑0 ∈ V)
17 ssid 3966 . . . . . . 7 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
1814, 5fexd 7177 . . . . . . . . 9 (𝜑𝐹 ∈ V)
19 suppimacnv 8105 . . . . . . . . 9 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2018, 15, 19sylancl 586 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2120sseq1d 3975 . . . . . . 7 (𝜑 → ((𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })) ↔ (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))))
2217, 21mpbiri 257 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
2314, 5, 16, 22gsumcllem 19685 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
2423oveq2d 7373 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘𝐴0 )))
2523oveq2d 7373 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
2613, 24, 253eqtr4d 2786 . . 3 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
2726ex 413 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
28 simprl 769 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
29 nnuz 12806 . . . . . . . 8 ℕ = (ℤ‘1)
3028, 29eleqtrdi 2848 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ (ℤ‘1))
3114adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴𝐵)
32 ffn 6668 . . . . . . . . . . . 12 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
33 dffn4 6762 . . . . . . . . . . . 12 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
3432, 33sylib 217 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
35 fof 6756 . . . . . . . . . . 11 (𝐹:𝐴onto→ran 𝐹𝐹:𝐴⟶ran 𝐹)
3631, 34, 353syl 18 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶ran 𝐹)
371adantr 481 . . . . . . . . . . . 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)
3938submacs 18637 . . . . . . . . . . . 12 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
40 acsmre 17532 . . . . . . . . . . . 12 ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
4137, 39, 403syl 18 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
42 eqid 2736 . . . . . . . . . . 11 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
4331frnd 6676 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹𝐵)
4441, 42, 43mrcssidd 17505 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
4536, 44fssd 6686 . . . . . . . . 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 6783 . . . . . . . . . . . 12 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
4746ad2antll 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 })))
48 cnvimass 6033 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
4948, 31fssdm 6688 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
50 f1ss 6744 . . . . . . . . . . 11 ((𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })) ∧ (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
5147, 49, 50syl2anc 584 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
52 f1f 6738 . . . . . . . . . 10 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
5351, 52syl 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 6692 . . . . . . . . 9 ((𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
5545, 53, 54syl2anc 584 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
5655ffvelcdmda 7035 . . . . . . 7 (((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) ∧ 𝑥 ∈ (1...(♯‘(𝐹 “ (V ∖ { 0 }))))) → ((𝐹𝑓)‘𝑥) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
5742mrccl 17491 . . . . . . . . . 10 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
5841, 43, 57syl2anc 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))
592oppgsubm 19143 . . . . . . . . 9 (SubMnd‘𝐺) = (SubMnd‘𝑂)
6058, 59eleqtrdi 2848 . . . . . . . 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 2736 . . . . . . . . . 10 (+g𝑂) = (+g𝑂)
6261submcl 18623 . . . . . . . . 9 ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂) ∧ 𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g𝑂)𝑦) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
63623expb 1120 . . . . . . . 8 ((((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂) ∧ (𝑥 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))) → (𝑥(+g𝑂)𝑦) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
6460, 63sylan 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(+g𝑂)𝑦) ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
65 eqid 2736 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
6665, 2, 61oppgplus 19127 . . . . . . . . 9 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝐺)𝑥)
67 gsumzoppg.c . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
6867adantr 481 . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . 14 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
7169, 42, 70cntzspan 19622 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
7237, 68, 71syl2anc 584 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
7370, 69submcmn2 19617 . . . . . . . . . . . . 13 (((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺) → ((𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
7458, 73syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd ↔ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ⊆ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))))
7572, 74mpbid 231 . . . . . . . . . . 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)))
7675sselda 3944 . . . . . . . . . 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)))
7765, 69cntzi 19109 . . . . . . . . . 10 ((𝑥 ∈ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
7876, 77sylan 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(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
7966, 78eqtr4id 2795 . . . . . . . 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(+g𝑂)𝑦) = (𝑥(+g𝐺)𝑦))
8079anasss 467 . . . . . . 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(+g𝑂)𝑦) = (𝑥(+g𝐺)𝑦))
8130, 56, 64, 80seqfeq4 13957 . . . . . 6 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (seq1((+g𝑂), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
822, 38oppgbas 19130 . . . . . . 7 𝐵 = (Base‘𝑂)
83 eqid 2736 . . . . . . 7 (Cntz‘𝑂) = (Cntz‘𝑂)
8437, 3syl 17 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑂 ∈ Mnd)
855adantr 481 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐴𝑉)
862, 69oppgcntz 19145 . . . . . . . 8 (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
8768, 86sseqtrdi 3994 . . . . . . 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 8108 . . . . . . . . . . 11 (𝐹 supp 0 ) ⊆ dom 𝐹
8920, 88eqsstrrdi 3999 . . . . . . . . . 10 (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
9014, 89fssdmd 6687 . . . . . . . . 9 (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
9190adantr 481 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
9247, 91, 50syl2anc 584 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
9321adantr 481 . . . . . . . . 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 }))))
9417, 93mpbiri 257 . . . . . . . 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 6791 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
96 forn 6759 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
9795, 96syl 17 . . . . . . . . . 10 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
9897sseq2d 3976 . . . . . . . . 9 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 }))))
9998ad2antll 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 }))))
10094, 99mpbird 256 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
101 eqid 2736 . . . . . . 7 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
10282, 7, 61, 83, 84, 85, 31, 87, 28, 92, 100, 101gsumval3 19684 . . . . . 6 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (seq1((+g𝑂), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
10322adantr 481 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
104103, 99mpbird 256 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
10538, 6, 65, 69, 37, 85, 31, 68, 28, 92, 104, 101gsumval3 19684 . . . . . 6 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐺 Σg 𝐹) = (seq1((+g𝐺), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
10681, 102, 1053eqtr4d 2786 . . . . 5 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
107106expr 457 . . . 4 ((𝜑 ∧ (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
108107exlimdv 1936 . . 3 ((𝜑 ∧ (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
109108expimpd 454 . 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 (𝜑𝐹 finSupp 0 )
111110fsuppimpd 9312 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
11220, 111eqeltrrd 2839 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
113 fz1f1o 15595 . . 3 ((𝐹 “ (V ∖ { 0 })) ∈ Fin → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
114112, 113syl 17 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
11527, 109, 114mpjaod 858 1 (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wex 1781  wcel 2106  Vcvv 3445  cdif 3907  wss 3910  c0 4282  {csn 4586   class class class wbr 5105  cmpt 5188  ccnv 5632  dom cdm 5633  ran crn 5634  cima 5636  ccom 5637   Fn wfn 6491  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357   supp csupp 8092  Fincfn 8883   finSupp cfsupp 9305  1c1 11052  cn 12153  cuz 12763  ...cfz 13424  seqcseq 13906  chash 14230  Basecbs 17083  s cress 17112  +gcplusg 17133  0gc0g 17321   Σg cgsu 17322  Moorecmre 17462  mrClscmrc 17463  ACScacs 17465  Mndcmnd 18556  SubMndcsubmnd 18600  Cntzccntz 19095  oppgcoppg 19123  CMndccmn 19562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-cntz 19097  df-oppg 19124  df-cmn 19564
This theorem is referenced by:  gsumzinv  19722
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