Metamath Proof Explorer

Theorem gsumpropd

Description: The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd etc. (Contributed by Stefan O'Rear, 1-Feb-2015) (Proof shortened by Mario Carneiro, 18-Sep-2015)

Ref Expression
Hypotheses gsumpropd.f ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
gsumpropd.g ⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
gsumpropd.h ⊢ ( 𝜑 → 𝐻 ∈ 𝑋 )
gsumpropd.b ⊢ ( 𝜑 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐻 ) )
gsumpropd.p ⊢ ( 𝜑 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝐻 ) )
Assertion gsumpropd ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐻 Σg 𝐹 ) )

Proof

Step Hyp Ref Expression
1 gsumpropd.f ⊢ ( 𝜑 → 𝐹 ∈ 𝑉 )
2 gsumpropd.g ⊢ ( 𝜑 → 𝐺 ∈ 𝑊 )
3 gsumpropd.h ⊢ ( 𝜑 → 𝐻 ∈ 𝑋 )
4 gsumpropd.b ⊢ ( 𝜑 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐻 ) )
5 gsumpropd.p ⊢ ( 𝜑 → ( +g ‘ 𝐺 ) = ( +g ‘ 𝐻 ) )
6 5 oveqd ⊢ ( 𝜑 → ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) )
7 6 eqeq1d ⊢ ( 𝜑 → ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ↔ ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ) )
8 5 oveqd ⊢ ( 𝜑 → ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) )
9 8 eqeq1d ⊢ ( 𝜑 → ( ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ↔ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) )
10 7 9 anbi12d ⊢ ( 𝜑 → ( ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) ↔ ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) ) )
11 4 10 raleqbidv ⊢ ( 𝜑 → ( ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) ↔ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) ) )
12 4 11 rabeqbidv ⊢ ( 𝜑 → { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } = { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } )
13 12 sseq2d ⊢ ( 𝜑 → ( ran 𝐹 ⊆ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ↔ ran 𝐹 ⊆ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) )
14 eqidd ⊢ ( 𝜑 → ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 ) )
15 5 oveqdr ⊢ ( ( 𝜑 ∧ ( 𝑎 ∈ ( Base ‘ 𝐺 ) ∧ 𝑏 ∈ ( Base ‘ 𝐺 ) ) ) → ( 𝑎 ( +g ‘ 𝐺 ) 𝑏 ) = ( 𝑎 ( +g ‘ 𝐻 ) 𝑏 ) )
16 14 4 15 grpidpropd ⊢ ( 𝜑 → ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐻 ) )
17 5 seqeq2d ⊢ ( 𝜑 → seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) = seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) )
18 17 fveq1d ⊢ ( 𝜑 → ( seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) ‘ 𝑛 ) = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) )
19 18 eqeq2d ⊢ ( 𝜑 → ( 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) ‘ 𝑛 ) ↔ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) )
20 19 anbi2d ⊢ ( 𝜑 → ( ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) ‘ 𝑛 ) ) ↔ ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) )
21 20 rexbidv ⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) ‘ 𝑛 ) ) ↔ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) )
22 21 exbidv ⊢ ( 𝜑 → ( ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) ‘ 𝑛 ) ) ↔ ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) )
23 22 iotabidv ⊢ ( 𝜑 → ( ℩ 𝑥 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) ‘ 𝑛 ) ) ) = ( ℩ 𝑥 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) )
24 12 difeq2d ⊢ ( 𝜑 → ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) = ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) )
25 24 imaeq2d ⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) )
26 25 fveq2d ⊢ ( 𝜑 → ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) = ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) )
27 26 oveq2d ⊢ ( 𝜑 → ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) = ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) )
28 27 f1oeq2d ⊢ ( 𝜑 → ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ↔ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) )
29 25 f1oeq3d ⊢ ( 𝜑 → ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ↔ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) )
30 28 29 bitrd ⊢ ( 𝜑 → ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ↔ 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) )
31 5 seqeq2d ⊢ ( 𝜑 → seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) = seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) )
32 31 26 fveq12d ⊢ ( 𝜑 → ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) )
33 32 eqeq2d ⊢ ( 𝜑 → ( 𝑥 = ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ↔ 𝑥 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) )
34 30 33 anbi12d ⊢ ( 𝜑 → ( ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ↔ ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) )
35 34 exbidv ⊢ ( 𝜑 → ( ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) )
36 35 iotabidv ⊢ ( 𝜑 → ( ℩ 𝑥 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) = ( ℩ 𝑥 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) )
37 23 36 ifeq12d ⊢ ( 𝜑 → if ( dom 𝐹 ∈ ran ... , ( ℩ 𝑥 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑥 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) ) = if ( dom 𝐹 ∈ ran ... , ( ℩ 𝑥 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑥 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) ) )
38 13 16 37 ifbieq12d ⊢ ( 𝜑 → if ( ran 𝐹 ⊆ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } , ( 0g ‘ 𝐺 ) , if ( dom 𝐹 ∈ ran ... , ( ℩ 𝑥 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑥 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) ) ) = if ( ran 𝐹 ⊆ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } , ( 0g ‘ 𝐻 ) , if ( dom 𝐹 ∈ ran ... , ( ℩ 𝑥 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑥 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) ) ) )
39 eqid ⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
40 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
41 eqid ⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 )
42 eqid ⊢ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } = { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) }
43 eqidd ⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) )
44 eqidd ⊢ ( 𝜑 → dom 𝐹 = dom 𝐹 )
45 39 40 41 42 43 2 1 44 gsumvalx ⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = if ( ran 𝐹 ⊆ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } , ( 0g ‘ 𝐺 ) , if ( dom 𝐹 ∈ ran ... , ( ℩ 𝑥 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐺 ) , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑥 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐺 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐺 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐺 ) ( ( 𝑠 ( +g ‘ 𝐺 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐺 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) ) ) )
46 eqid ⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 )
47 eqid ⊢ ( 0g ‘ 𝐻 ) = ( 0g ‘ 𝐻 )
48 eqid ⊢ ( +g ‘ 𝐻 ) = ( +g ‘ 𝐻 )
49 eqid ⊢ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } = { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) }
50 eqidd ⊢ ( 𝜑 → ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) )
51 46 47 48 49 50 3 1 44 gsumvalx ⊢ ( 𝜑 → ( 𝐻 Σg 𝐹 ) = if ( ran 𝐹 ⊆ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } , ( 0g ‘ 𝐻 ) , if ( dom 𝐹 ∈ ran ... , ( ℩ 𝑥 ∃ 𝑚 ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ( dom 𝐹 = ( 𝑚 ... 𝑛 ) ∧ 𝑥 = ( seq 𝑚 ( ( +g ‘ 𝐻 ) , 𝐹 ) ‘ 𝑛 ) ) ) , ( ℩ 𝑥 ∃ 𝑓 ( 𝑓 : ( 1 ... ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) –1-1-onto→ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ∧ 𝑥 = ( seq 1 ( ( +g ‘ 𝐻 ) , ( 𝐹 ∘ 𝑓 ) ) ‘ ( ♯ ‘ ( ◡ 𝐹 “ ( V ∖ { 𝑠 ∈ ( Base ‘ 𝐻 ) ∣ ∀ 𝑡 ∈ ( Base ‘ 𝐻 ) ( ( 𝑠 ( +g ‘ 𝐻 ) 𝑡 ) = 𝑡 ∧ ( 𝑡 ( +g ‘ 𝐻 ) 𝑠 ) = 𝑡 ) } ) ) ) ) ) ) ) ) )
52 38 45 51 3eqtr4d ⊢ ( 𝜑 → ( 𝐺 Σg 𝐹 ) = ( 𝐻 Σg 𝐹 ) )