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Theorem gsumress 18596
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 oveq1 7410 . . . . . . . . . 10 (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
21eqeq1d 2735 . . . . . . . . 9 (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
32ovanraleqv 7427 . . . . . . . 8 (𝑦 = 0 → (∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
4 gsumress.s . . . . . . . . 9 (𝜑𝑆𝐵)
5 gsumress.z . . . . . . . . 9 (𝜑0𝑆)
64, 5sseldd 3981 . . . . . . . 8 (𝜑0𝐵)
7 gsumress.c . . . . . . . . 9 ((𝜑𝑥𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
87ralrimiva 3147 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
93, 6, 8elrabd 3683 . . . . . . 7 (𝜑0 ∈ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
109snssd 4810 . . . . . 6 (𝜑 → { 0 } ⊆ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
11 gsumress.g . . . . . . . 8 (𝜑𝐺𝑉)
12 gsumress.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
13 eqid 2733 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
14 gsumress.o . . . . . . . . 9 + = (+g𝐺)
15 eqid 2733 . . . . . . . . 9 {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}
1612, 13, 14, 15mgmidsssn0 18586 . . . . . . . 8 (𝐺𝑉 → {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g𝐺)})
1711, 16syl 17 . . . . . . 7 (𝜑 → {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ {(0g𝐺)})
1817, 9sseldd 3981 . . . . . . . . 9 (𝜑0 ∈ {(0g𝐺)})
19 elsni 4643 . . . . . . . . 9 ( 0 ∈ {(0g𝐺)} → 0 = (0g𝐺))
2018, 19syl 17 . . . . . . . 8 (𝜑0 = (0g𝐺))
2120sneqd 4638 . . . . . . 7 (𝜑 → { 0 } = {(0g𝐺)})
2217, 21sseqtrrd 4021 . . . . . 6 (𝜑 → {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ { 0 })
2310, 22eqssd 3997 . . . . 5 (𝜑 → { 0 } = {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
242ovanraleqv 7427 . . . . . . . . 9 (𝑦 = 0 → (∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
254sselda 3980 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑥𝐵)
2625, 7syldan 592 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
2726ralrimiva 3147 . . . . . . . . 9 (𝜑 → ∀𝑥𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
2824, 5, 27elrabd 3683 . . . . . . . 8 (𝜑0 ∈ {𝑦𝑆 ∣ ∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
29 gsumress.h . . . . . . . . . . 11 𝐻 = (𝐺s 𝑆)
3029, 12ressbas2 17177 . . . . . . . . . 10 (𝑆𝐵𝑆 = (Base‘𝐻))
314, 30syl 17 . . . . . . . . 9 (𝜑𝑆 = (Base‘𝐻))
32 fvex 6900 . . . . . . . . . . . . . . 15 (Base‘𝐻) ∈ V
3331, 32eqeltrdi 2842 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ V)
3429, 14ressplusg 17230 . . . . . . . . . . . . . 14 (𝑆 ∈ V → + = (+g𝐻))
3533, 34syl 17 . . . . . . . . . . . . 13 (𝜑+ = (+g𝐻))
3635oveqd 7420 . . . . . . . . . . . 12 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐻)𝑥))
3736eqeq1d 2735 . . . . . . . . . . 11 (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g𝐻)𝑥) = 𝑥))
3835oveqd 7420 . . . . . . . . . . . 12 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐻)𝑦))
3938eqeq1d 2735 . . . . . . . . . . 11 (𝜑 → ((𝑥 + 𝑦) = 𝑥 ↔ (𝑥(+g𝐻)𝑦) = 𝑥))
4037, 39anbi12d 632 . . . . . . . . . 10 (𝜑 → (((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)))
4131, 40raleqbidv 3343 . . . . . . . . 9 (𝜑 → (∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)))
4231, 41rabeqbidv 3450 . . . . . . . 8 (𝜑 → {𝑦𝑆 ∣ ∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
4328, 42eleqtrd 2836 . . . . . . 7 (𝜑0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
4443snssd 4810 . . . . . 6 (𝜑 → { 0 } ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
4529ovexi 7437 . . . . . . . . 9 𝐻 ∈ V
4645a1i 11 . . . . . . . 8 (𝜑𝐻 ∈ V)
47 eqid 2733 . . . . . . . . 9 (Base‘𝐻) = (Base‘𝐻)
48 eqid 2733 . . . . . . . . 9 (0g𝐻) = (0g𝐻)
49 eqid 2733 . . . . . . . . 9 (+g𝐻) = (+g𝐻)
50 eqid 2733 . . . . . . . . 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)}
5147, 48, 49, 50mgmidsssn0 18586 . . . . . . . 8 (𝐻 ∈ V → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)} ⊆ {(0g𝐻)})
5246, 51syl 17 . . . . . . 7 (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)} ⊆ {(0g𝐻)})
5352, 43sseldd 3981 . . . . . . . . 9 (𝜑0 ∈ {(0g𝐻)})
54 elsni 4643 . . . . . . . . 9 ( 0 ∈ {(0g𝐻)} → 0 = (0g𝐻))
5553, 54syl 17 . . . . . . . 8 (𝜑0 = (0g𝐻))
5655sneqd 4638 . . . . . . 7 (𝜑 → { 0 } = {(0g𝐻)})
5752, 56sseqtrrd 4021 . . . . . 6 (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)} ⊆ { 0 })
5844, 57eqssd 3997 . . . . 5 (𝜑 → { 0 } = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
5923, 58eqtr3d 2775 . . . 4 (𝜑 → {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
6059sseq2d 4012 . . 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)}))
6120, 55eqtr3d 2775 . . 3 (𝜑 → (0g𝐺) = (0g𝐻))
6235seqeq2d 13968 . . . . . . . . . 10 (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g𝐻), 𝐹))
6362fveq1d 6889 . . . . . . . . 9 (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
6463eqeq2d 2744 . . . . . . . 8 (𝜑 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
6564anbi2d 630 . . . . . . 7 (𝜑 → ((𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6665rexbidv 3179 . . . . . 6 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6766exbidv 1925 . . . . 5 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6867iotabidv 6523 . . . 4 (𝜑 → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6935seqeq2d 13968 . . . . . . . . 9 (𝜑 → seq1( + , (𝐹𝑓)) = seq1((+g𝐻), (𝐹𝑓)))
7069fveq1d 6889 . . . . . . . 8 (𝜑 → (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
7170eqeq2d 2744 . . . . . . 7 (𝜑 → (𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))) ↔ 𝑧 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 }))))))
7271anbi2d 630 . . . . . 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 })))))))
7372exbidv 1925 . . . . 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 })))))))
7473iotabidv 6523 . . . 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 })))))))
7568, 74ifeq12d 4547 . . 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 }))))))))
7660, 61, 75ifbieq12d 4554 . 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 })))))))))
7723difeq2d 4120 . . . 4 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}))
7877imaeq2d 6056 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})))
79 gsumress.a . . 3 (𝜑𝐴𝑋)
80 gsumress.f . . . 4 (𝜑𝐹:𝐴𝑆)
8180, 4fssd 6731 . . 3 (𝜑𝐹:𝐴𝐵)
8212, 13, 14, 15, 78, 11, 79, 81gsumval 18591 . 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 })))))))))
8358difeq2d 4120 . . . 4 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)}))
8483imaeq2d 6056 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})))
8531feq3d 6700 . . . 4 (𝜑 → (𝐹:𝐴𝑆𝐹:𝐴⟶(Base‘𝐻)))
8680, 85mpbid 231 . . 3 (𝜑𝐹:𝐴⟶(Base‘𝐻))
8747, 48, 49, 50, 84, 46, 79, 86gsumval 18591 . 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 })))))))))
8876, 82, 873eqtr4d 2783 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  cdif 3943  wss 3946  ifcif 4526  {csn 4626  ccnv 5673  ran crn 5675  cima 5677  ccom 5678  cio 6489  wf 6535  1-1-ontowf1o 6538  cfv 6539  (class class class)co 7403  1c1 11106  cuz 12817  ...cfz 13479  seqcseq 13961  chash 14285  Basecbs 17139  s cress 17168  +gcplusg 17192  0gc0g 17380   Σg cgsu 17381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-om 7850  df-2nd 7970  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-nn 12208  df-2 12270  df-seq 13962  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17140  df-ress 17169  df-plusg 17205  df-0g 17382  df-gsum 17383
This theorem is referenced by:  gsumsubm  18711  regsumfsum  20997  regsumsupp  21158  frlmgsum  21310  imasdsf1olem  23860  gsumsubg  32175  esumpfinvallem  33009  sge0tsms  45030  aacllem  47749
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