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Theorem gsumress 18695
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 7438 . . . . . . . . . 10 (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
21eqeq1d 2739 . . . . . . . . 9 (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
32ovanraleqv 7455 . . . . . . . 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 3984 . . . . . . . 8 (𝜑0𝐵)
7 gsumress.c . . . . . . . . 9 ((𝜑𝑥𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
87ralrimiva 3146 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
93, 6, 8elrabd 3694 . . . . . . 7 (𝜑0 ∈ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
109snssd 4809 . . . . . 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 2737 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
14 gsumress.o . . . . . . . . 9 + = (+g𝐺)
15 eqid 2737 . . . . . . . . 9 {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}
1612, 13, 14, 15mgmidsssn0 18685 . . . . . . . 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 3984 . . . . . . . . 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 4001 . . . . 5 (𝜑 → { 0 } = {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
242ovanraleqv 7455 . . . . . . . . 9 (𝑦 = 0 → (∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
254sselda 3983 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑥𝐵)
2625, 7syldan 591 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
2726ralrimiva 3146 . . . . . . . . 9 (𝜑 → ∀𝑥𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
2824, 5, 27elrabd 3694 . . . . . . . 8 (𝜑0 ∈ {𝑦𝑆 ∣ ∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
29 gsumress.h . . . . . . . . . . 11 𝐻 = (𝐺s 𝑆)
3029, 12ressbas2 17283 . . . . . . . . . 10 (𝑆𝐵𝑆 = (Base‘𝐻))
314, 30syl 17 . . . . . . . . 9 (𝜑𝑆 = (Base‘𝐻))
32 fvex 6919 . . . . . . . . . . . . . . 15 (Base‘𝐻) ∈ V
3331, 32eqeltrdi 2849 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ V)
3429, 14ressplusg 17334 . . . . . . . . . . . . . 14 (𝑆 ∈ V → + = (+g𝐻))
3533, 34syl 17 . . . . . . . . . . . . 13 (𝜑+ = (+g𝐻))
3635oveqd 7448 . . . . . . . . . . . 12 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐻)𝑥))
3736eqeq1d 2739 . . . . . . . . . . 11 (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g𝐻)𝑥) = 𝑥))
3835oveqd 7448 . . . . . . . . . . . 12 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐻)𝑦))
3938eqeq1d 2739 . . . . . . . . . . 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 3346 . . . . . . . . 9 (𝜑 → (∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)))
4231, 41rabeqbidv 3455 . . . . . . . 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 2843 . . . . . . 7 (𝜑0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
4443snssd 4809 . . . . . 6 (𝜑 → { 0 } ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
4529ovexi 7465 . . . . . . . . 9 𝐻 ∈ V
4645a1i 11 . . . . . . . 8 (𝜑𝐻 ∈ V)
47 eqid 2737 . . . . . . . . 9 (Base‘𝐻) = (Base‘𝐻)
48 eqid 2737 . . . . . . . . 9 (0g𝐻) = (0g𝐻)
49 eqid 2737 . . . . . . . . 9 (+g𝐻) = (+g𝐻)
50 eqid 2737 . . . . . . . . 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 18685 . . . . . . . 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 3984 . . . . . . . . 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 4001 . . . . 5 (𝜑 → { 0 } = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
5923, 58eqtr3d 2779 . . . 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 4016 . . 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 2779 . . 3 (𝜑 → (0g𝐺) = (0g𝐻))
6235seqeq2d 14049 . . . . . . . . . 10 (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g𝐻), 𝐹))
6362fveq1d 6908 . . . . . . . . 9 (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
6463eqeq2d 2748 . . . . . . . 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 1921 . . . . 5 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6867iotabidv 6545 . . . 4 (𝜑 → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6935seqeq2d 14049 . . . . . . . . 9 (𝜑 → seq1( + , (𝐹𝑓)) = seq1((+g𝐻), (𝐹𝑓)))
7069fveq1d 6908 . . . . . . . 8 (𝜑 → (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
7170eqeq2d 2748 . . . . . . 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 1921 . . . . 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 6545 . . . 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 4126 . . . 4 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}))
7877imaeq2d 6078 . . 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 6753 . . 3 (𝜑𝐹:𝐴𝐵)
8212, 13, 14, 15, 78, 11, 79, 81gsumval 18690 . 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 4126 . . . 4 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)}))
8483imaeq2d 6078 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})))
8531feq3d 6723 . . . 4 (𝜑 → (𝐹:𝐴𝑆𝐹:𝐴⟶(Base‘𝐻)))
8680, 85mpbid 232 . . 3 (𝜑𝐹:𝐴⟶(Base‘𝐻))
8747, 48, 49, 50, 84, 46, 79, 86gsumval 18690 . 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 2787 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  cdif 3948  wss 3951  ifcif 4525  {csn 4626  ccnv 5684  ran crn 5686  cima 5688  ccom 5689  cio 6512  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  1c1 11156  cuz 12878  ...cfz 13547  seqcseq 14042  chash 14369  Basecbs 17247  s cress 17274  +gcplusg 17297  0gc0g 17484   Σg cgsu 17485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-seq 14043  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487
This theorem is referenced by:  gsumsubm  18848  regsumfsum  21453  regsumsupp  21640  frlmgsum  21792  imasdsf1olem  24383  gsumsubg  33049  gsumzrsum  33062  esumpfinvallem  34075  sge0tsms  46395  aacllem  49320
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