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Theorem gsumzoppg 19061
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 18478 . . . . . . . 8 (𝐺 ∈ Mnd → 𝑂 ∈ Mnd)
41, 3syl 17 . . . . . . 7 (𝜑𝑂 ∈ Mnd)
5 gsumzoppg.a . . . . . . 7 (𝜑𝐴𝑉)
6 gsumzoppg.0 . . . . . . . . 9 0 = (0g𝐺)
72, 6oppgid 18480 . . . . . . . 8 0 = (0g𝑂)
87gsumz 17996 . . . . . . 7 ((𝑂 ∈ Mnd ∧ 𝐴𝑉) → (𝑂 Σg (𝑘𝐴0 )) = 0 )
94, 5, 8syl2anc 587 . . . . . 6 (𝜑 → (𝑂 Σg (𝑘𝐴0 )) = 0 )
106gsumz 17996 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
111, 5, 10syl2anc 587 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
129, 11eqtr4d 2839 . . . . 5 (𝜑 → (𝑂 Σg (𝑘𝐴0 )) = (𝐺 Σg (𝑘𝐴0 )))
1312adantr 484 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg (𝑘𝐴0 )) = (𝐺 Σg (𝑘𝐴0 )))
14 gsumzoppg.f . . . . . 6 (𝜑𝐹:𝐴𝐵)
156fvexi 6663 . . . . . . 7 0 ∈ V
1615a1i 11 . . . . . 6 (𝜑0 ∈ V)
17 ssid 3940 . . . . . . 7 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
18 fex 6970 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
1914, 5, 18syl2anc 587 . . . . . . . . 9 (𝜑𝐹 ∈ V)
20 suppimacnv 7828 . . . . . . . . 9 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2119, 15, 20sylancl 589 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2221sseq1d 3949 . . . . . . 7 (𝜑 → ((𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })) ↔ (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))))
2317, 22mpbiri 261 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
2414, 5, 16, 23gsumcllem 19025 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
2524oveq2d 7155 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘𝐴0 )))
2624oveq2d 7155 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
2713, 25, 263eqtr4d 2846 . . 3 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
2827ex 416 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
29 simprl 770 . . . . . . . 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)
3129, 30eleqtrdi 2903 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ (ℤ‘1))
3214adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴𝐵)
33 ffn 6491 . . . . . . . . . . . 12 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
34 dffn4 6575 . . . . . . . . . . . 12 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
3533, 34sylib 221 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
36 fof 6569 . . . . . . . . . . 11 (𝐹:𝐴onto→ran 𝐹𝐹:𝐴⟶ran 𝐹)
3732, 35, 363syl 18 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶ran 𝐹)
381adantr 484 . . . . . . . . . . . 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)
4039submacs 17987 . . . . . . . . . . . 12 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
41 acsmre 16919 . . . . . . . . . . . 12 ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
4238, 40, 413syl 18 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
43 eqid 2801 . . . . . . . . . . 11 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
4432frnd 6498 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹𝐵)
4542, 43, 44mrcssidd 16892 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
4637, 45fssd 6506 . . . . . . . . 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 6593 . . . . . . . . . . . 12 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
4847ad2antll 728 . . . . . . . . . . 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 5920 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
5049, 32fssdm 6508 . . . . . . . . . . 11 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
51 f1ss 6559 . . . . . . . . . . 11 ((𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })) ∧ (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
5248, 50, 51syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
53 f1f 6553 . . . . . . . . . 10 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
5452, 53syl 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 6509 . . . . . . . . 9 ((𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹𝑓):(1...(♯‘(𝐹 “ (V ∖ { 0 }))))⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
5646, 54, 55syl2anc 587 . . . . . . . 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))
5756ffvelrnda 6832 . . . . . . 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))
5843mrccl 16878 . . . . . . . . . 10 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
5942, 44, 58syl2anc 587 . . . . . . . . 9 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
602oppgsubm 18486 . . . . . . . . 9 (SubMnd‘𝐺) = (SubMnd‘𝑂)
6159, 60eleqtrdi 2903 . . . . . . . 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 2801 . . . . . . . . . 10 (+g𝑂) = (+g𝑂)
6362submcl 17973 . . . . . . . . 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))
64633expb 1117 . . . . . . . 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))
6561, 64sylan 583 . . . . . . 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))
66 eqid 2801 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
6766, 2, 62oppgplus 18473 . . . . . . . . 9 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝐺)𝑥)
68 gsumzoppg.c . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
6968adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
70 gsumzoppg.z . . . . . . . . . . . . . 14 𝑍 = (Cntz‘𝐺)
71 eqid 2801 . . . . . . . . . . . . . 14 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
7270, 43, 71cntzspan 18961 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
7338, 69, 72syl2anc 587 . . . . . . . . . . . 12 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
7471, 70submcmn2 18956 . . . . . . . . . . . . 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))))
7559, 74syl 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))))
7673, 75mpbid 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)))
7776sselda 3918 . . . . . . . . . 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)))
7866, 70cntzi 18455 . . . . . . . . . 10 ((𝑥 ∈ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
7977, 78sylan 583 . . . . . . . . 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))
8067, 79eqtr4id 2855 . . . . . . . 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))
8180anasss 470 . . . . . . 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))
8231, 57, 65, 81seqfeq4 13419 . . . . . 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 })))))
832, 39oppgbas 18475 . . . . . . 7 𝐵 = (Base‘𝑂)
84 eqid 2801 . . . . . . 7 (Cntz‘𝑂) = (Cntz‘𝑂)
8538, 3syl 17 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑂 ∈ Mnd)
865adantr 484 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐴𝑉)
872, 70oppgcntz 18488 . . . . . . . 8 (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
8869, 87sseqtrdi 3968 . . . . . . 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 7830 . . . . . . . . . . 11 (𝐹 supp 0 ) ⊆ dom 𝐹
9021, 89eqsstrrdi 3973 . . . . . . . . . 10 (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
9114, 90fssdmd 6507 . . . . . . . . 9 (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
9291adantr 484 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
9348, 92, 51syl2anc 587 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
9422adantr 484 . . . . . . . . 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 }))))
9517, 94mpbiri 261 . . . . . . . 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 6601 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
97 forn 6572 . . . . . . . . . . 11 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
9896, 97syl 17 . . . . . . . . . 10 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
9998sseq2d 3950 . . . . . . . . 9 (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 }))))
10099ad2antll 728 . . . . . . . 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 }))))
10195, 100mpbird 260 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
102 eqid 2801 . . . . . . 7 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
10383, 7, 62, 84, 85, 86, 32, 88, 29, 93, 101, 102gsumval3 19024 . . . . . 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 })))))
10423adantr 484 . . . . . . . 8 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
105104, 100mpbird 260 . . . . . . 7 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
10639, 6, 66, 70, 38, 86, 32, 69, 29, 93, 105, 102gsumval3 19024 . . . . . 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 })))))
10782, 103, 1063eqtr4d 2846 . . . . 5 ((𝜑 ∧ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
108107expr 460 . . . 4 ((𝜑 ∧ (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
109108exlimdv 1934 . . 3 ((𝜑 ∧ (♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
110109expimpd 457 . 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 (𝜑𝐹 finSupp 0 )
112111fsuppimpd 8828 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
11321, 112eqeltrrd 2894 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
114 fz1f1o 15063 . . 3 ((𝐹 “ (V ∖ { 0 })) ∈ Fin → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
115113, 114syl 17 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((♯‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
11628, 110, 115mpjaod 857 1 (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wex 1781  wcel 2112  Vcvv 3444  cdif 3881  wss 3884  c0 4246  {csn 4528   class class class wbr 5033  cmpt 5113  ccnv 5522  dom cdm 5523  ran crn 5524  cima 5526  ccom 5527   Fn wfn 6323  wf 6324  1-1wf1 6325  ontowfo 6326  1-1-ontowf1o 6327  cfv 6328  (class class class)co 7139   supp csupp 7817  Fincfn 8496   finSupp cfsupp 8821  1c1 10531  cn 11629  cuz 12235  ...cfz 12889  seqcseq 13368  chash 13690  Basecbs 16479  s cress 16480  +gcplusg 16561  0gc0g 16709   Σg cgsu 16710  Moorecmre 16849  mrClscmrc 16850  ACScacs 16852  Mndcmnd 17907  SubMndcsubmnd 17951  Cntzccntz 18441  oppgcoppg 18469  CMndccmn 18902
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-0g 16711  df-gsum 16712  df-mre 16853  df-mrc 16854  df-acs 16856  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-submnd 17953  df-cntz 18443  df-oppg 18470  df-cmn 18904
This theorem is referenced by:  gsumzinv  19062
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