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Theorem gsumress 17483
Description: The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither 𝐺 nor 𝐻 need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
gsumress.b 𝐵 = (Base‘𝐺)
gsumress.o + = (+g𝐺)
gsumress.h 𝐻 = (𝐺s 𝑆)
gsumress.g (𝜑𝐺𝑉)
gsumress.a (𝜑𝐴𝑋)
gsumress.s (𝜑𝑆𝐵)
gsumress.f (𝜑𝐹:𝐴𝑆)
gsumress.z (𝜑0𝑆)
gsumress.c ((𝜑𝑥𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
Assertion
Ref Expression
gsumress (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝜑,𝑥   𝑥,𝑆   𝑥,𝐻   𝑥, +   𝑥, 0
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑋(𝑥)

Proof of Theorem gsumress
Dummy variables 𝑓 𝑚 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 gsumress.s . . . . . . . . 9 (𝜑𝑆𝐵)
2 gsumress.z . . . . . . . . 9 (𝜑0𝑆)
31, 2sseldd 3751 . . . . . . . 8 (𝜑0𝐵)
4 gsumress.c . . . . . . . . 9 ((𝜑𝑥𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
54ralrimiva 3114 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
6 oveq1 6799 . . . . . . . . . . . 12 (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
76eqeq1d 2772 . . . . . . . . . . 11 (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
8 oveq2 6800 . . . . . . . . . . . 12 (𝑦 = 0 → (𝑥 + 𝑦) = (𝑥 + 0 ))
98eqeq1d 2772 . . . . . . . . . . 11 (𝑦 = 0 → ((𝑥 + 𝑦) = 𝑥 ↔ (𝑥 + 0 ) = 𝑥))
107, 9anbi12d 608 . . . . . . . . . 10 (𝑦 = 0 → (((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
1110ralbidv 3134 . . . . . . . . 9 (𝑦 = 0 → (∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
1211elrab 3513 . . . . . . . 8 ( 0 ∈ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ↔ ( 0𝐵 ∧ ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
133, 5, 12sylanbrc 564 . . . . . . 7 (𝜑0 ∈ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
1413snssd 4473 . . . . . 6 (𝜑 → { 0 } ⊆ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
15 gsumress.g . . . . . . . 8 (𝜑𝐺𝑉)
16 gsumress.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
17 eqid 2770 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
18 gsumress.o . . . . . . . . 9 + = (+g𝐺)
19 eqid 2770 . . . . . . . . 9 {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}
2016, 17, 18, 19mgmidsssn0 17476 . . . . . . . 8 (𝐺𝑉 → {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g𝐺)})
2115, 20syl 17 . . . . . . 7 (𝜑 → {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g𝐺)})
2221, 13sseldd 3751 . . . . . . . . 9 (𝜑0 ∈ {(0g𝐺)})
23 elsni 4331 . . . . . . . . 9 ( 0 ∈ {(0g𝐺)} → 0 = (0g𝐺))
2422, 23syl 17 . . . . . . . 8 (𝜑0 = (0g𝐺))
2524sneqd 4326 . . . . . . 7 (𝜑 → { 0 } = {(0g𝐺)})
2621, 25sseqtr4d 3789 . . . . . 6 (𝜑 → {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ { 0 })
2714, 26eqssd 3767 . . . . 5 (𝜑 → { 0 } = {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
281sselda 3750 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑥𝐵)
2928, 4syldan 571 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
3029ralrimiva 3114 . . . . . . . . 9 (𝜑 → ∀𝑥𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
3110ralbidv 3134 . . . . . . . . . 10 (𝑦 = 0 → (∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
3231elrab 3513 . . . . . . . . 9 ( 0 ∈ {𝑦𝑆 ∣ ∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ↔ ( 0𝑆 ∧ ∀𝑥𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
332, 30, 32sylanbrc 564 . . . . . . . 8 (𝜑0 ∈ {𝑦𝑆 ∣ ∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
34 gsumress.h . . . . . . . . . . 11 𝐻 = (𝐺s 𝑆)
3534, 16ressbas2 16137 . . . . . . . . . 10 (𝑆𝐵𝑆 = (Base‘𝐻))
361, 35syl 17 . . . . . . . . 9 (𝜑𝑆 = (Base‘𝐻))
37 fvex 6342 . . . . . . . . . . . . . . 15 (Base‘𝐻) ∈ V
3836, 37syl6eqel 2857 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ V)
3934, 18ressplusg 16200 . . . . . . . . . . . . . 14 (𝑆 ∈ V → + = (+g𝐻))
4038, 39syl 17 . . . . . . . . . . . . 13 (𝜑+ = (+g𝐻))
4140oveqd 6809 . . . . . . . . . . . 12 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐻)𝑥))
4241eqeq1d 2772 . . . . . . . . . . 11 (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g𝐻)𝑥) = 𝑥))
4340oveqd 6809 . . . . . . . . . . . 12 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐻)𝑦))
4443eqeq1d 2772 . . . . . . . . . . 11 (𝜑 → ((𝑥 + 𝑦) = 𝑥 ↔ (𝑥(+g𝐻)𝑦) = 𝑥))
4542, 44anbi12d 608 . . . . . . . . . 10 (𝜑 → (((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)))
4636, 45raleqbidv 3300 . . . . . . . . 9 (𝜑 → (∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)))
4736, 46rabeqbidv 3344 . . . . . . . 8 (𝜑 → {𝑦𝑆 ∣ ∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
4833, 47eleqtrd 2851 . . . . . . 7 (𝜑0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
4948snssd 4473 . . . . . 6 (𝜑 → { 0 } ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
50 ovex 6822 . . . . . . . . . 10 (𝐺s 𝑆) ∈ V
5134, 50eqeltri 2845 . . . . . . . . 9 𝐻 ∈ V
5251a1i 11 . . . . . . . 8 (𝜑𝐻 ∈ V)
53 eqid 2770 . . . . . . . . 9 (Base‘𝐻) = (Base‘𝐻)
54 eqid 2770 . . . . . . . . 9 (0g𝐻) = (0g𝐻)
55 eqid 2770 . . . . . . . . 9 (+g𝐻) = (+g𝐻)
56 eqid 2770 . . . . . . . . 9 {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)} = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)}
5753, 54, 55, 56mgmidsssn0 17476 . . . . . . . 8 (𝐻 ∈ V → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)} ⊆ {(0g𝐻)})
5852, 57syl 17 . . . . . . 7 (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)} ⊆ {(0g𝐻)})
5958, 48sseldd 3751 . . . . . . . . 9 (𝜑0 ∈ {(0g𝐻)})
60 elsni 4331 . . . . . . . . 9 ( 0 ∈ {(0g𝐻)} → 0 = (0g𝐻))
6159, 60syl 17 . . . . . . . 8 (𝜑0 = (0g𝐻))
6261sneqd 4326 . . . . . . 7 (𝜑 → { 0 } = {(0g𝐻)})
6358, 62sseqtr4d 3789 . . . . . 6 (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)} ⊆ { 0 })
6449, 63eqssd 3767 . . . . 5 (𝜑 → { 0 } = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
6527, 64eqtr3d 2806 . . . 4 (𝜑 → {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
6665sseq2d 3780 . . 3 (𝜑 → (ran 𝐹 ⊆ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ↔ ran 𝐹 ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)}))
6724, 61eqtr3d 2806 . . 3 (𝜑 → (0g𝐺) = (0g𝐻))
6840seqeq2d 13014 . . . . . . . . . 10 (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g𝐻), 𝐹))
6968fveq1d 6334 . . . . . . . . 9 (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
7069eqeq2d 2780 . . . . . . . 8 (𝜑 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
7170anbi2d 606 . . . . . . 7 (𝜑 → ((𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
7271rexbidv 3199 . . . . . 6 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
7372exbidv 2001 . . . . 5 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
7473iotabidv 6015 . . . 4 (𝜑 → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
7540seqeq2d 13014 . . . . . . . . 9 (𝜑 → seq1( + , (𝐹𝑓)) = seq1((+g𝐻), (𝐹𝑓)))
7675fveq1d 6334 . . . . . . . 8 (𝜑 → (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
7776eqeq2d 2780 . . . . . . 7 (𝜑 → (𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))) ↔ 𝑧 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 }))))))
7877anbi2d 606 . . . . . 6 (𝜑 → ((𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 }))))) ↔ (𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))))
7978exbidv 2001 . . . . 5 (𝜑 → (∃𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 }))))) ↔ ∃𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))))
8079iotabidv 6015 . . . 4 (𝜑 → (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))) = (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))))
8174, 80ifeq12d 4243 . . 3 (𝜑 → if(𝐴 ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 }))))))) = if(𝐴 ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 }))))))))
8266, 67, 81ifbieq12d 4250 . 2 (𝜑 → if(ran 𝐹 ⊆ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}, (0g𝐺), if(𝐴 ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))))) = if(ran 𝐹 ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)}, (0g𝐻), if(𝐴 ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))))))
8327difeq2d 3877 . . . 4 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}))
8483imaeq2d 5607 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})))
85 gsumress.a . . 3 (𝜑𝐴𝑋)
86 gsumress.f . . . 4 (𝜑𝐹:𝐴𝑆)
8786, 1fssd 6197 . . 3 (𝜑𝐹:𝐴𝐵)
8816, 17, 18, 19, 84, 15, 85, 87gsumval 17478 . 2 (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}, (0g𝐺), if(𝐴 ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))))))
8964difeq2d 3877 . . . 4 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)}))
9089imaeq2d 5607 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})))
9136feq3d 6172 . . . 4 (𝜑 → (𝐹:𝐴𝑆𝐹:𝐴⟶(Base‘𝐻)))
9286, 91mpbid 222 . . 3 (𝜑𝐹:𝐴⟶(Base‘𝐻))
9353, 54, 55, 56, 90, 52, 85, 92gsumval 17478 . 2 (𝜑 → (𝐻 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)}, (0g𝐻), if(𝐴 ∈ ran ..., (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))), (℩𝑧𝑓(𝑓:(1...(♯‘(𝐹 “ (V ∖ { 0 }))))–1-1-onto→(𝐹 “ (V ∖ { 0 })) ∧ 𝑧 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))))))
9482, 88, 933eqtr4d 2814 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wex 1851  wcel 2144  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  cdif 3718  wss 3721  ifcif 4223  {csn 4314  ccnv 5248  ran crn 5250  cima 5252  ccom 5253  cio 5992  wf 6027  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  1c1 10138  cuz 11887  ...cfz 12532  seqcseq 13007  chash 13320  Basecbs 16063  s cress 16064  +gcplusg 16148  0gc0g 16307   Σg cgsu 16308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-seq 13008  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-0g 16309  df-gsum 16310
This theorem is referenced by:  gsumsubm  17580  regsumfsum  20028  regsumsupp  20184  frlmgsum  20327  imasdsf1olem  22397  esumpfinvallem  30470  sge0tsms  41108  aacllem  43068
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