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Theorem gsumress 17880
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 7152 . . . . . . . . . 10 (𝑦 = 0 → (𝑦 + 𝑥) = ( 0 + 𝑥))
21eqeq1d 2820 . . . . . . . . 9 (𝑦 = 0 → ((𝑦 + 𝑥) = 𝑥 ↔ ( 0 + 𝑥) = 𝑥))
32ovanraleqv 7169 . . . . . . . 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 3965 . . . . . . . 8 (𝜑0𝐵)
7 gsumress.c . . . . . . . . 9 ((𝜑𝑥𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
87ralrimiva 3179 . . . . . . . 8 (𝜑 → ∀𝑥𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
93, 6, 8elrabd 3679 . . . . . . 7 (𝜑0 ∈ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
109snssd 4734 . . . . . 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 2818 . . . . . . . . 9 (0g𝐺) = (0g𝐺)
14 gsumress.o . . . . . . . . 9 + = (+g𝐺)
15 eqid 2818 . . . . . . . . 9 {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} = {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}
1612, 13, 14, 15mgmidsssn0 17870 . . . . . . . 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 3965 . . . . . . . . 9 (𝜑0 ∈ {(0g𝐺)})
19 elsni 4574 . . . . . . . . 9 ( 0 ∈ {(0g𝐺)} → 0 = (0g𝐺))
2018, 19syl 17 . . . . . . . 8 (𝜑0 = (0g𝐺))
2120sneqd 4569 . . . . . . 7 (𝜑 → { 0 } = {(0g𝐺)})
2217, 21sseqtrrd 4005 . . . . . 6 (𝜑 → {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)} ⊆ { 0 })
2310, 22eqssd 3981 . . . . 5 (𝜑 → { 0 } = {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
242ovanraleqv 7169 . . . . . . . . 9 (𝑦 = 0 → (∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
254sselda 3964 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑥𝐵)
2625, 7syldan 591 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
2726ralrimiva 3179 . . . . . . . . 9 (𝜑 → ∀𝑥𝑆 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
2824, 5, 27elrabd 3679 . . . . . . . 8 (𝜑0 ∈ {𝑦𝑆 ∣ ∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)})
29 gsumress.h . . . . . . . . . . 11 𝐻 = (𝐺s 𝑆)
3029, 12ressbas2 16543 . . . . . . . . . 10 (𝑆𝐵𝑆 = (Base‘𝐻))
314, 30syl 17 . . . . . . . . 9 (𝜑𝑆 = (Base‘𝐻))
32 fvex 6676 . . . . . . . . . . . . . . 15 (Base‘𝐻) ∈ V
3331, 32syl6eqel 2918 . . . . . . . . . . . . . 14 (𝜑𝑆 ∈ V)
3429, 14ressplusg 16600 . . . . . . . . . . . . . 14 (𝑆 ∈ V → + = (+g𝐻))
3533, 34syl 17 . . . . . . . . . . . . 13 (𝜑+ = (+g𝐻))
3635oveqd 7162 . . . . . . . . . . . 12 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐻)𝑥))
3736eqeq1d 2820 . . . . . . . . . . 11 (𝜑 → ((𝑦 + 𝑥) = 𝑥 ↔ (𝑦(+g𝐻)𝑥) = 𝑥))
3835oveqd 7162 . . . . . . . . . . . 12 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐻)𝑦))
3938eqeq1d 2820 . . . . . . . . . . 11 (𝜑 → ((𝑥 + 𝑦) = 𝑥 ↔ (𝑥(+g𝐻)𝑦) = 𝑥))
4037, 39anbi12d 630 . . . . . . . . . 10 (𝜑 → (((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)))
4131, 40raleqbidv 3399 . . . . . . . . 9 (𝜑 → (∀𝑥𝑆 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)))
4231, 41rabeqbidv 3483 . . . . . . . 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 2912 . . . . . . 7 (𝜑0 ∈ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
4443snssd 4734 . . . . . 6 (𝜑 → { 0 } ⊆ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
4529ovexi 7179 . . . . . . . . 9 𝐻 ∈ V
4645a1i 11 . . . . . . . 8 (𝜑𝐻 ∈ V)
47 eqid 2818 . . . . . . . . 9 (Base‘𝐻) = (Base‘𝐻)
48 eqid 2818 . . . . . . . . 9 (0g𝐻) = (0g𝐻)
49 eqid 2818 . . . . . . . . 9 (+g𝐻) = (+g𝐻)
50 eqid 2818 . . . . . . . . 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 17870 . . . . . . . 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 3965 . . . . . . . . 9 (𝜑0 ∈ {(0g𝐻)})
54 elsni 4574 . . . . . . . . 9 ( 0 ∈ {(0g𝐻)} → 0 = (0g𝐻))
5553, 54syl 17 . . . . . . . 8 (𝜑0 = (0g𝐻))
5655sneqd 4569 . . . . . . 7 (𝜑 → { 0 } = {(0g𝐻)})
5752, 56sseqtrrd 4005 . . . . . 6 (𝜑 → {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)} ⊆ { 0 })
5844, 57eqssd 3981 . . . . 5 (𝜑 → { 0 } = {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})
5923, 58eqtr3d 2855 . . . 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 3996 . . 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 2855 . . 3 (𝜑 → (0g𝐺) = (0g𝐻))
6235seqeq2d 13364 . . . . . . . . . 10 (𝜑 → seq𝑚( + , 𝐹) = seq𝑚((+g𝐻), 𝐹))
6362fveq1d 6665 . . . . . . . . 9 (𝜑 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑚((+g𝐻), 𝐹)‘𝑛))
6463eqeq2d 2829 . . . . . . . 8 (𝜑 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛)))
6564anbi2d 628 . . . . . . 7 (𝜑 → ((𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ (𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6665rexbidv 3294 . . . . . 6 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6766exbidv 1913 . . . . 5 (𝜑 → (∃𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6867iotabidv 6332 . . . 4 (𝜑 → (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (℩𝑧𝑚𝑛 ∈ (ℤ𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚((+g𝐻), 𝐹)‘𝑛))))
6935seqeq2d 13364 . . . . . . . . 9 (𝜑 → seq1( + , (𝐹𝑓)) = seq1((+g𝐻), (𝐹𝑓)))
7069fveq1d 6665 . . . . . . . 8 (𝜑 → (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))) = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))))
7170eqeq2d 2829 . . . . . . 7 (𝜑 → (𝑧 = (seq1( + , (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 })))) ↔ 𝑧 = (seq1((+g𝐻), (𝐹𝑓))‘(♯‘(𝐹 “ (V ∖ { 0 }))))))
7271anbi2d 628 . . . . . 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 1913 . . . . 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 6332 . . . 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 4483 . . 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 4490 . 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 4096 . . . 4 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦𝐵 ∣ ∀𝑥𝐵 ((𝑦 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑦) = 𝑥)}))
7877imaeq2d 5922 . . 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 6521 . . 3 (𝜑𝐹:𝐴𝐵)
8212, 13, 14, 15, 78, 11, 79, 81gsumval 17875 . 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 4096 . . . 4 (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)}))
8483imaeq2d 5922 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) = (𝐹 “ (V ∖ {𝑦 ∈ (Base‘𝐻) ∣ ∀𝑥 ∈ (Base‘𝐻)((𝑦(+g𝐻)𝑥) = 𝑥 ∧ (𝑥(+g𝐻)𝑦) = 𝑥)})))
8531feq3d 6494 . . . 4 (𝜑 → (𝐹:𝐴𝑆𝐹:𝐴⟶(Base‘𝐻)))
8680, 85mpbid 233 . . 3 (𝜑𝐹:𝐴⟶(Base‘𝐻))
8747, 48, 49, 50, 84, 46, 79, 86gsumval 17875 . 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 2863 1 (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cdif 3930  wss 3933  ifcif 4463  {csn 4557  ccnv 5547  ran crn 5549  cima 5551  ccom 5552  cio 6305  wf 6344  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  1c1 10526  cuz 12231  ...cfz 12880  seqcseq 13357  chash 13678  Basecbs 16471  s cress 16472  +gcplusg 16553  0gc0g 16701   Σg cgsu 16702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-seq 13358  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-gsum 16704
This theorem is referenced by:  gsumsubm  17987  regsumfsum  20541  regsumsupp  20694  frlmgsum  20844  imasdsf1olem  22910  gsumsubg  30611  esumpfinvallem  31232  sge0tsms  42539  aacllem  44830
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