MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gsumzoppg Structured version   Visualization version   GIF version

Theorem gsumzoppg 18167
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 17607 . . . . . . . 8 (𝐺 ∈ Mnd → 𝑂 ∈ Mnd)
41, 3syl 17 . . . . . . 7 (𝜑𝑂 ∈ Mnd)
5 gsumzoppg.a . . . . . . 7 (𝜑𝐴𝑉)
6 gsumzoppg.0 . . . . . . . . 9 0 = (0g𝐺)
72, 6oppgid 17609 . . . . . . . 8 0 = (0g𝑂)
87gsumz 17197 . . . . . . 7 ((𝑂 ∈ Mnd ∧ 𝐴𝑉) → (𝑂 Σg (𝑘𝐴0 )) = 0 )
94, 5, 8syl2anc 691 . . . . . 6 (𝜑 → (𝑂 Σg (𝑘𝐴0 )) = 0 )
106gsumz 17197 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
111, 5, 10syl2anc 691 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
129, 11eqtr4d 2647 . . . . 5 (𝜑 → (𝑂 Σg (𝑘𝐴0 )) = (𝐺 Σg (𝑘𝐴0 )))
1312adantr 480 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg (𝑘𝐴0 )) = (𝐺 Σg (𝑘𝐴0 )))
14 gsumzoppg.f . . . . . 6 (𝜑𝐹:𝐴𝐵)
15 fvex 6113 . . . . . . . 8 (0g𝐺) ∈ V
166, 15eqeltri 2684 . . . . . . 7 0 ∈ V
1716a1i 11 . . . . . 6 (𝜑0 ∈ V)
18 ssid 3587 . . . . . . 7 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
19 fex 6394 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
2014, 5, 19syl2anc 691 . . . . . . . . 9 (𝜑𝐹 ∈ V)
21 suppimacnv 7193 . . . . . . . . 9 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2220, 16, 21sylancl 693 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2322sseq1d 3595 . . . . . . 7 (𝜑 → ((𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })) ↔ (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))))
2418, 23mpbiri 247 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
2514, 5, 17, 24gsumcllem 18132 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
2625oveq2d 6565 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘𝐴0 )))
2725oveq2d 6565 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
2813, 26, 273eqtr4d 2654 . . 3 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
2928ex 449 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
30 simprl 790 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
31 nnuz 11599 . . . . . . . 8 ℕ = (ℤ‘1)
3230, 31syl6eleq 2698 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (#‘(𝐹 “ (V ∖ { 0 }))) ∈ (ℤ‘1))
3314adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴𝐵)
34 ffn 5958 . . . . . . . . . . . 12 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
35 dffn4 6034 . . . . . . . . . . . 12 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
3634, 35sylib 207 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
37 fof 6028 . . . . . . . . . . 11 (𝐹:𝐴onto→ran 𝐹𝐹:𝐴⟶ran 𝐹)
3833, 36, 373syl 18 . . . . . . . . . 10 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶ran 𝐹)
391adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐺 ∈ Mnd)
40 gsumzoppg.b . . . . . . . . . . . . 13 𝐵 = (Base‘𝐺)
4140submacs 17188 . . . . . . . . . . . 12 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
42 acsmre 16136 . . . . . . . . . . . 12 ((SubMnd‘𝐺) ∈ (ACS‘𝐵) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
4339, 41, 423syl 18 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (SubMnd‘𝐺) ∈ (Moore‘𝐵))
44 eqid 2610 . . . . . . . . . . 11 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
45 frn 5966 . . . . . . . . . . . 12 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
4633, 45syl 17 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹𝐵)
4743, 44, 46mrcssidd 16108 . . . . . . . . . 10 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
4838, 47fssd 5970 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
49 f1of1 6049 . . . . . . . . . . . 12 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
5049ad2antll 761 . . . . . . . . . . 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 })))
51 cnvimass 5404 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
52 fdm 5964 . . . . . . . . . . . . 13 (𝐹:𝐴𝐵 → dom 𝐹 = 𝐴)
5333, 52syl 17 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → dom 𝐹 = 𝐴)
5451, 53syl5sseq 3616 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
55 f1ss 6019 . . . . . . . . . . 11 ((𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })) ∧ (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
5650, 54, 55syl2anc 691 . . . . . . . . . 10 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
57 f1f 6014 . . . . . . . . . 10 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
5856, 57syl 17 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴)
59 fco 5971 . . . . . . . . 9 ((𝐹:𝐴⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶𝐴) → (𝐹𝑓):(1...(#‘(𝐹 “ (V ∖ { 0 }))))⟶((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
6048, 58, 59syl2anc 691 . . . . . . . 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))
6160ffvelrnda 6267 . . . . . . 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))
6244mrccl 16094 . . . . . . . . . 10 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
6343, 46, 62syl2anc 691 . . . . . . . . 9 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
642oppgsubm 17615 . . . . . . . . 9 (SubMnd‘𝐺) = (SubMnd‘𝑂)
6563, 64syl6eleq 2698 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝑂))
66 eqid 2610 . . . . . . . . . 10 (+g𝑂) = (+g𝑂)
6766submcl 17176 . . . . . . . . 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))
68673expb 1258 . . . . . . . 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))
6965, 68sylan 487 . . . . . . 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))
70 gsumzoppg.c . . . . . . . . . . . . . 14 (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
7170adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
72 gsumzoppg.z . . . . . . . . . . . . . 14 𝑍 = (Cntz‘𝐺)
73 eqid 2610 . . . . . . . . . . . . . 14 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
7472, 44, 73cntzspan 18070 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
7539, 71, 74syl2anc 691 . . . . . . . . . . . 12 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
7673, 72submcmn2 18067 . . . . . . . . . . . . 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))))
7763, 76syl 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))))
7875, 77mpbid 221 . . . . . . . . . . 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)))
7978sselda 3568 . . . . . . . . . 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)))
80 eqid 2610 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
8180, 72cntzi 17585 . . . . . . . . . 10 ((𝑥 ∈ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
8279, 81sylan 487 . . . . . . . . 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))
8380, 2, 66oppgplus 17602 . . . . . . . . 9 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝐺)𝑥)
8482, 83syl6reqr 2663 . . . . . . . 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))
8584anasss 677 . . . . . . 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))
8632, 61, 69, 85seqfeq4 12712 . . . . . 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 })))))
872, 40oppgbas 17604 . . . . . . 7 𝐵 = (Base‘𝑂)
88 eqid 2610 . . . . . . 7 (Cntz‘𝑂) = (Cntz‘𝑂)
8939, 3syl 17 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑂 ∈ Mnd)
905adantr 480 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐴𝑉)
912, 72oppgcntz 17617 . . . . . . . 8 (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
9271, 91syl6sseq 3614 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → ran 𝐹 ⊆ ((Cntz‘𝑂)‘ran 𝐹))
93 suppssdm 7195 . . . . . . . . . . . . . . 15 (𝐹 supp 0 ) ⊆ dom 𝐹
9422, 93syl6eqssr 3619 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
9594adantl 481 . . . . . . . . . . . . 13 ((dom 𝐹 = 𝐴𝜑) → (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
96 eqcom 2617 . . . . . . . . . . . . . . 15 (dom 𝐹 = 𝐴𝐴 = dom 𝐹)
9796biimpi 205 . . . . . . . . . . . . . 14 (dom 𝐹 = 𝐴𝐴 = dom 𝐹)
9897adantr 480 . . . . . . . . . . . . 13 ((dom 𝐹 = 𝐴𝜑) → 𝐴 = dom 𝐹)
9995, 98sseqtr4d 3605 . . . . . . . . . . . 12 ((dom 𝐹 = 𝐴𝜑) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
10099ex 449 . . . . . . . . . . 11 (dom 𝐹 = 𝐴 → (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴))
10152, 100syl 17 . . . . . . . . . 10 (𝐹:𝐴𝐵 → (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴))
10214, 101mpcom 37 . . . . . . . . 9 (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
103102adantr 480 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
10450, 103, 55syl2anc 691 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1𝐴)
10523adantr 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 }))))
10618, 105mpbiri 247 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
107 f1ofo 6057 . . . . . . . . . . 11 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
108 forn 6031 . . . . . . . . . . 11 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
109107, 108syl 17 . . . . . . . . . 10 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ran 𝑓 = (𝐹 “ (V ∖ { 0 })))
110109sseq2d 3596 . . . . . . . . 9 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 }))))
111110ad2antll 761 . . . . . . . 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 }))))
112106, 111mpbird 246 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
113 eqid 2610 . . . . . . 7 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
11487, 7, 66, 88, 89, 90, 33, 92, 30, 104, 112, 113gsumval3 18131 . . . . . 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 })))))
11524adantr 480 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
116115, 111mpbird 246 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
11740, 6, 80, 72, 39, 90, 33, 71, 30, 104, 116, 113gsumval3 18131 . . . . . 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 })))))
11886, 114, 1173eqtr4d 2654 . . . . 5 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
119118expr 641 . . . 4 ((𝜑 ∧ (#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
120119exlimdv 1848 . . 3 ((𝜑 ∧ (#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
121120expimpd 627 . 2 (𝜑 → (((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 }))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
122 gsumzoppg.n . . . . 5 (𝜑𝐹 finSupp 0 )
123122fsuppimpd 8165 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
12422, 123eqeltrrd 2689 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
125 fz1f1o 14288 . . 3 ((𝐹 “ (V ∖ { 0 })) ∈ Fin → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
126124, 125syl 17 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ ∨ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))))
12729, 121, 126mpjaod 395 1 (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wo 382  wa 383   = wceq 1475  wex 1695  wcel 1977  Vcvv 3173  cdif 3537  wss 3540  c0 3874  {csn 4125   class class class wbr 4583  cmpt 4643  ccnv 5037  dom cdm 5038  ran crn 5039  cima 5041  ccom 5042   Fn wfn 5799  wf 5800  1-1wf1 5801  ontowfo 5802  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549   supp csupp 7182  Fincfn 7841   finSupp cfsupp 8158  1c1 9816  cn 10897  cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979  Basecbs 15695  s cress 15696  +gcplusg 15768  0gc0g 15923   Σg cgsu 15924  Moorecmre 16065  mrClscmrc 16066  ACScacs 16068  Mndcmnd 17117  SubMndcsubmnd 17157  Cntzccntz 17571  oppgcoppg 17598  CMndccmn 18016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-cntz 17573  df-oppg 17599  df-cmn 18018
This theorem is referenced by:  gsumzinv  18168
  Copyright terms: Public domain W3C validator