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Theorem gsumzoppg 18260
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 17700 . . . . . . . 8 (𝐺 ∈ Mnd → 𝑂 ∈ Mnd)
41, 3syl 17 . . . . . . 7 (𝜑𝑂 ∈ Mnd)
5 gsumzoppg.a . . . . . . 7 (𝜑𝐴𝑉)
6 gsumzoppg.0 . . . . . . . . 9 0 = (0g𝐺)
72, 6oppgid 17702 . . . . . . . 8 0 = (0g𝑂)
87gsumz 17290 . . . . . . 7 ((𝑂 ∈ Mnd ∧ 𝐴𝑉) → (𝑂 Σg (𝑘𝐴0 )) = 0 )
94, 5, 8syl2anc 692 . . . . . 6 (𝜑 → (𝑂 Σg (𝑘𝐴0 )) = 0 )
106gsumz 17290 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝐴𝑉) → (𝐺 Σg (𝑘𝐴0 )) = 0 )
111, 5, 10syl2anc 692 . . . . . 6 (𝜑 → (𝐺 Σg (𝑘𝐴0 )) = 0 )
129, 11eqtr4d 2663 . . . . 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)
15 fvex 6160 . . . . . . . 8 (0g𝐺) ∈ V
166, 15eqeltri 2700 . . . . . . 7 0 ∈ V
1716a1i 11 . . . . . 6 (𝜑0 ∈ V)
18 ssid 3608 . . . . . . 7 (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 “ (V ∖ { 0 }))
19 fex 6445 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝐴𝑉) → 𝐹 ∈ V)
2014, 5, 19syl2anc 692 . . . . . . . . 9 (𝜑𝐹 ∈ V)
21 suppimacnv 7252 . . . . . . . . 9 ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2220, 16, 21sylancl 693 . . . . . . . 8 (𝜑 → (𝐹 supp 0 ) = (𝐹 “ (V ∖ { 0 })))
2322sseq1d 3616 . . . . . . 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 18225 . . . . 5 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → 𝐹 = (𝑘𝐴0 ))
2625oveq2d 6621 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝑂 Σg (𝑘𝐴0 )))
2725oveq2d 6621 . . . 4 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝐺 Σg 𝐹) = (𝐺 Σg (𝑘𝐴0 )))
2813, 26, 273eqtr4d 2670 . . 3 ((𝜑 ∧ (𝐹 “ (V ∖ { 0 })) = ∅) → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
2928ex 450 . 2 (𝜑 → ((𝐹 “ (V ∖ { 0 })) = ∅ → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹)))
30 simprl 793 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ)
31 nnuz 11667 . . . . . . . 8 ℕ = (ℤ‘1)
3230, 31syl6eleq 2714 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (#‘(𝐹 “ (V ∖ { 0 }))) ∈ (ℤ‘1))
3314adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐹:𝐴𝐵)
34 ffn 6004 . . . . . . . . . . . 12 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
35 dffn4 6080 . . . . . . . . . . . 12 (𝐹 Fn 𝐴𝐹:𝐴onto→ran 𝐹)
3634, 35sylib 208 . . . . . . . . . . 11 (𝐹:𝐴𝐵𝐹:𝐴onto→ran 𝐹)
37 fof 6074 . . . . . . . . . . 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 481 . . . . . . . . . . . 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 17281 . . . . . . . . . . . 12 (𝐺 ∈ Mnd → (SubMnd‘𝐺) ∈ (ACS‘𝐵))
42 acsmre 16229 . . . . . . . . . . . 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 2626 . . . . . . . . . . 11 (mrCls‘(SubMnd‘𝐺)) = (mrCls‘(SubMnd‘𝐺))
45 frn 6012 . . . . . . . . . . . 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 16201 . . . . . . . . . 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 6016 . . . . . . . . 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 6095 . . . . . . . . . . . 12 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1→(𝐹 “ (V ∖ { 0 })))
5049ad2antll 764 . . . . . . . . . . 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 5448 . . . . . . . . . . . 12 (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹
52 fdm 6010 . . . . . . . . . . . . 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 3637 . . . . . . . . . . 11 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
55 f1ss 6065 . . . . . . . . . . 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 692 . . . . . . . . . 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 6060 . . . . . . . . . 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 6017 . . . . . . . . 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 692 . . . . . . . 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 6316 . . . . . . 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 16187 . . . . . . . . . 10 (((SubMnd‘𝐺) ∈ (Moore‘𝐵) ∧ ran 𝐹𝐵) → ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹) ∈ (SubMnd‘𝐺))
6343, 46, 62syl2anc 692 . . . . . . . . 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 17708 . . . . . . . . 9 (SubMnd‘𝐺) = (SubMnd‘𝑂)
6563, 64syl6eleq 2714 . . . . . . . 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 2626 . . . . . . . . . 10 (+g𝑂) = (+g𝑂)
6766submcl 17269 . . . . . . . . 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 1263 . . . . . . . 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 488 . . . . . . 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 481 . . . . . . . . . . . . 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 2626 . . . . . . . . . . . . . 14 (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) = (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹))
7472, 44, 73cntzspan 18163 . . . . . . . . . . . . 13 ((𝐺 ∈ Mnd ∧ ran 𝐹 ⊆ (𝑍‘ran 𝐹)) → (𝐺s ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∈ CMnd)
7539, 71, 74syl2anc 692 . . . . . . . . . . . 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 18160 . . . . . . . . . . . . 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 3588 . . . . . . . . . 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 2626 . . . . . . . . . . 11 (+g𝐺) = (+g𝐺)
8180, 72cntzi 17678 . . . . . . . . . 10 ((𝑥 ∈ (𝑍‘((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) ∧ 𝑦 ∈ ((mrCls‘(SubMnd‘𝐺))‘ran 𝐹)) → (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))
8279, 81sylan 488 . . . . . . . . 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 17695 . . . . . . . . 9 (𝑥(+g𝑂)𝑦) = (𝑦(+g𝐺)𝑥)
8482, 83syl6reqr 2679 . . . . . . . 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 678 . . . . . . 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 12787 . . . . . 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 17697 . . . . . . 7 𝐵 = (Base‘𝑂)
88 eqid 2626 . . . . . . 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 481 . . . . . . 7 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → 𝐴𝑉)
912, 72oppgcntz 17710 . . . . . . . 8 (𝑍‘ran 𝐹) = ((Cntz‘𝑂)‘ran 𝐹)
9271, 91syl6sseq 3635 . . . . . . 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 7254 . . . . . . . . . . . . . . 15 (𝐹 supp 0 ) ⊆ dom 𝐹
9422, 93syl6eqssr 3640 . . . . . . . . . . . . . 14 (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
9594adantl 482 . . . . . . . . . . . . 13 ((dom 𝐹 = 𝐴𝜑) → (𝐹 “ (V ∖ { 0 })) ⊆ dom 𝐹)
96 eqcom 2633 . . . . . . . . . . . . . . 15 (dom 𝐹 = 𝐴𝐴 = dom 𝐹)
9796biimpi 206 . . . . . . . . . . . . . 14 (dom 𝐹 = 𝐴𝐴 = dom 𝐹)
9897adantr 481 . . . . . . . . . . . . 13 ((dom 𝐹 = 𝐴𝜑) → 𝐴 = dom 𝐹)
9995, 98sseqtr4d 3626 . . . . . . . . . . . 12 ((dom 𝐹 = 𝐴𝜑) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
10099ex 450 . . . . . . . . . . 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 481 . . . . . . . 8 ((𝜑 ∧ ((#‘(𝐹 “ (V ∖ { 0 }))) ∈ ℕ ∧ 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })))) → (𝐹 “ (V ∖ { 0 })) ⊆ 𝐴)
10450, 103, 55syl2anc 692 . . . . . . 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 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 }))))
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 6103 . . . . . . . . . . 11 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → 𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–onto→(𝐹 “ (V ∖ { 0 })))
108 forn 6077 . . . . . . . . . . 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 3617 . . . . . . . . 9 (𝑓:(1...(#‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) → ((𝐹 supp 0 ) ⊆ ran 𝑓 ↔ (𝐹 supp 0 ) ⊆ (𝐹 “ (V ∖ { 0 }))))
111110ad2antll 764 . . . . . . . 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 2626 . . . . . . 7 ((𝐹𝑓) supp 0 ) = ((𝐹𝑓) supp 0 )
11487, 7, 66, 88, 89, 90, 33, 92, 30, 104, 112, 113gsumval3 18224 . . . . . 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 481 . . . . . . . 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 18224 . . . . . 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 2670 . . . . 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 1863 . . 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 8227 . . . 4 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
12422, 123eqeltrrd 2705 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ∈ Fin)
125 fz1f1o 14369 . . 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 396 1 (𝜑 → (𝑂 Σg 𝐹) = (𝐺 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1992  Vcvv 3191  cdif 3557  wss 3560  c0 3896  {csn 4153   class class class wbr 4618  cmpt 4678  ccnv 5078  dom cdm 5079  ran crn 5080  cima 5082  ccom 5083   Fn wfn 5845  wf 5846  1-1wf1 5847  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605   supp csupp 7241  Fincfn 7900   finSupp cfsupp 8220  1c1 9882  cn 10965  cuz 11631  ...cfz 12265  seqcseq 12738  #chash 13054  Basecbs 15776  s cress 15777  +gcplusg 15857  0gc0g 16016   Σg cgsu 16017  Moorecmre 16158  mrClscmrc 16159  ACScacs 16161  Mndcmnd 17210  SubMndcsubmnd 17250  Cntzccntz 17664  oppgcoppg 17691  CMndccmn 18109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-iin 4493  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-supp 7242  df-tpos 7298  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-oadd 7510  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-fsupp 8221  df-oi 8360  df-card 8710  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-n0 11238  df-z 11323  df-uz 11632  df-fz 12266  df-fzo 12404  df-seq 12739  df-hash 13055  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-0g 16018  df-gsum 16019  df-mre 16162  df-mrc 16163  df-acs 16165  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-submnd 17252  df-cntz 17666  df-oppg 17692  df-cmn 18111
This theorem is referenced by:  gsumzinv  18261
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