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Theorem gsumzoppg 18390
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 17830 . . . . . . . 8 (𝐺 ∈ Mnd → 𝑂 ∈ Mnd)
41, 3syl 17 . . . . . . 7 (𝜑𝑂 ∈ Mnd)
5 gsumzoppg.a . . . . . . 7 (𝜑𝐴𝑉)
6 gsumzoppg.0 . . . . . . . . 9 0 = (0g𝐺)
72, 6oppgid 17832 . . . . . . . 8 0 = (0g𝑂)
87gsumz 17421 . . . . . . 7 ((𝑂 ∈ Mnd ∧ 𝐴𝑉) → (𝑂 Σg (𝑘𝐴0 )) = 0 )
94, 5, 8syl2anc 694 . . . . . 6 (𝜑 → (𝑂 Σg (𝑘𝐴0 )) = 0 )
106gsumz 17421 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
111, 5, 10syl2anc 694 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
129, 11eqtr4d 2688 . . . . 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 6239 . . . . . . . 8 (0g𝐺) ∈ V
166, 15eqeltri 2726 . . . . . . 7 0 ∈ V
1716a1i 11 . . . . . 6 (𝜑0 ∈ V)
18 ssid 3657 . . . . . . 7 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
19 fex 6530 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
2014, 5, 19syl2anc 694 . . . . . . . . 9 (𝜑𝐹 ∈ V)
21 suppimacnv 7351 . . . . . . . . 9 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2220, 16, 21sylancl 695 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2322sseq1d 3665 . . . . . . 7 (𝜑 → ((𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })) ↔ (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))))
2418, 23mpbiri 248 . . . . . 6 (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 })))
2514, 5, 17, 24gsumcllem 18355 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
2625oveq2d 6706 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘𝐴0 )))
2725oveq2d 6706 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
2813, 26, 273eqtr4d 2695 . . 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 809 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
31 nnuz 11761 . . . . . . . 8 ℕ = (ℤ‘1)
3230, 31syl6eleq 2740 . . . . . . 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 6083 . . . . . . . . . . . 12 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
35 dffn4 6159 . . . . . . . . . . . 12 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
3634, 35sylib 208 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
37 fof 6153 . . . . . . . . . . 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 17412 . . . . . . . . . . . 12 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
42 acsmre 16360 . . . . . . . . . . . 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 2651 . . . . . . . . . . 11 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
45 frn 6091 . . . . . . . . . . . 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 16332 . . . . . . . . . 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 6095 . . . . . . . . 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 6174 . . . . . . . . . . . 12 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
5049ad2antll 765 . . . . . . . . . . 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 5520 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
52 fdm 6089 . . . . . . . . . . . . 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 3686 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
55 f1ss 6144 . . . . . . . . . . 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 694 . . . . . . . . . 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 6139 . . . . . . . . . 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 6096 . . . . . . . . 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 694 . . . . . . . 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 6399 . . . . . . 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 16318 . . . . . . . . . 10 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
6343, 46, 62syl2anc 694 . . . . . . . . 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 17838 . . . . . . . . 9 (SubMnd‘𝐺) = (SubMnd‘𝑂)
6563, 64syl6eleq 2740 . . . . . . . 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 2651 . . . . . . . . . 10 (+g𝑂) = (+g𝑂)
6766submcl 17400 . . . . . . . . 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 1285 . . . . . . . 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 2651 . . . . . . . . . . . . . 14 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
7472, 44, 73cntzspan 18293 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
7539, 71, 74syl2anc 694 . . . . . . . . . . . 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 18290 . . . . . . . . . . . . 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 222 . . . . . . . . . . 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 3636 . . . . . . . . . 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 2651 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
8180, 72cntzi 17808 . . . . . . . . . 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 17825 . . . . . . . . 9 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝐺)𝑥)
8482, 83syl6reqr 2704 . . . . . . . 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 680 . . . . . . 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 12890 . . . . . 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 17827 . . . . . . 7 𝐵 = (Base‘𝑂)
88 eqid 2651 . . . . . . 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 17840 . . . . . . . 8 (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
9271, 91syl6sseq 3684 . . . . . . 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 7353 . . . . . . . . . . . . . . 15 (𝐹 supp 0 ) ⊆ dom 𝐹
9422, 93syl6eqssr 3689 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
9594adantl 481 . . . . . . . . . . . . 13 ((dom 𝐹 = 𝐴𝜑) → (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
96 eqcom 2658 . . . . . . . . . . . . . . 15 (dom 𝐹 = 𝐴𝐴 = dom 𝐹)
9796biimpi 206 . . . . . . . . . . . . . 14 (dom 𝐹 = 𝐴𝐴 = dom 𝐹)
9897adantr 480 . . . . . . . . . . . . 13 ((dom 𝐹 = 𝐴𝜑) → 𝐴 = dom 𝐹)
9995, 98sseqtr4d 3675 . . . . . . . . . . . 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 38 . . . . . . . . 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 694 . . . . . . 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 248 . . . . . . . 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 6182 . . . . . . . . . . 11 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
108 forn 6156 . . . . . . . . . . 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 3666 . . . . . . . . 9 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 }))))
111110ad2antll 765 . . . . . . . 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 247 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 supp 0 ) ⊆ ran 𝑓)
113 eqid 2651 . . . . . . 7 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
11487, 7, 66, 88, 89, 90, 33, 92, 30, 104, 112, 113gsumval3 18354 . . . . . 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 247 . . . . . . 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 18354 . . . . . 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 2695 . . . . 5 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
119118expr 642 . . . 4 ((𝜑 ∧ (#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ) → (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
120119exlimdv 1901 . . 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 628 . 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 8323 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
12422, 123eqeltrrd 2731 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
125 fz1f1o 14485 . . 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 196  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  cdif 3604  wss 3607  c0 3948  {csn 4210   class class class wbr 4685  cmpt 4762  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147   Fn wfn 5921  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690   supp csupp 7340  Fincfn 7997   finSupp cfsupp 8316  1c1 9975  cn 11058  cuz 11725  ...cfz 12364  seqcseq 12841  #chash 13157  Basecbs 15904  s cress 15905  +gcplusg 15988  0gc0g 16147   Σg cgsu 16148  Moorecmre 16289  mrClscmrc 16290  ACScacs 16292  Mndcmnd 17341  SubMndcsubmnd 17381  Cntzccntz 17794  oppgcoppg 17821  CMndccmn 18239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-cntz 17796  df-oppg 17822  df-cmn 18241
This theorem is referenced by:  gsumzinv  18391
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