Metamath Proof Explorer

Theorem pj1ghm

Description: The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016)

Ref Expression
Hypotheses pj1eu.a ⊢ + = ( +g ‘ 𝐺 )
pj1eu.s ⊢ ⊕ = ( LSSum ‘ 𝐺 )
pj1eu.o ⊢ 0 = ( 0g ‘ 𝐺 )
pj1eu.z ⊢ 𝑍 = ( Cntz ‘ 𝐺 )
pj1eu.2 ⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
pj1eu.3 ⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
pj1eu.4 ⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = { 0 } )
pj1eu.5 ⊢ ( 𝜑 → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) )
pj1f.p ⊢ 𝑃 = ( proj1 ‘ 𝐺 )
Assertion pj1ghm ( 𝜑 → ( 𝑇 𝑃 𝑈 ) ∈ ( ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) GrpHom 𝐺 ) )

Proof

Step Hyp Ref Expression
1 pj1eu.a ⊢ + = ( +g ‘ 𝐺 )
2 pj1eu.s ⊢ ⊕ = ( LSSum ‘ 𝐺 )
3 pj1eu.o ⊢ 0 = ( 0g ‘ 𝐺 )
4 pj1eu.z ⊢ 𝑍 = ( Cntz ‘ 𝐺 )
5 pj1eu.2 ⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
6 pj1eu.3 ⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
7 pj1eu.4 ⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = { 0 } )
8 pj1eu.5 ⊢ ( 𝜑 → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) )
9 pj1f.p ⊢ 𝑃 = ( proj1 ‘ 𝐺 )
10 eqid ⊢ ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) = ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) )
11 eqid ⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
12 ovex ⊢ ( 𝑇 ⊕ 𝑈 ) ∈ V
13 eqid ⊢ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) = ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) )
14 13 1 ressplusg ⊢ ( ( 𝑇 ⊕ 𝑈 ) ∈ V → + = ( +g ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) )
15 12 14 ax-mp ⊢ + = ( +g ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) )
16 2 4 lsmsubg ⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( 𝑇 ⊕ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) )
17 5 6 8 16 syl3anc ⊢ ( 𝜑 → ( 𝑇 ⊕ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) )
18 13 subggrp ⊢ ( ( 𝑇 ⊕ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ∈ Grp )
19 17 18 syl ⊢ ( 𝜑 → ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ∈ Grp )
20 subgrcl ⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 𝐺 ∈ Grp )
21 5 20 syl ⊢ ( 𝜑 → 𝐺 ∈ Grp )
22 1 2 3 4 5 6 7 8 9 pj1f ⊢ ( 𝜑 → ( 𝑇 𝑃 𝑈 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑇 )
23 11 subgss ⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 𝑇 ⊆ ( Base ‘ 𝐺 ) )
24 5 23 syl ⊢ ( 𝜑 → 𝑇 ⊆ ( Base ‘ 𝐺 ) )
25 22 24 fssd ⊢ ( 𝜑 → ( 𝑇 𝑃 𝑈 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ ( Base ‘ 𝐺 ) )
26 13 subgbas ⊢ ( ( 𝑇 ⊕ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) → ( 𝑇 ⊕ 𝑈 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) )
27 17 26 syl ⊢ ( 𝜑 → ( 𝑇 ⊕ 𝑈 ) = ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) )
28 27 feq2d ⊢ ( 𝜑 → ( ( 𝑇 𝑃 𝑈 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ ( Base ‘ 𝐺 ) ↔ ( 𝑇 𝑃 𝑈 ) : ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ⟶ ( Base ‘ 𝐺 ) ) )
29 25 28 mpbid ⊢ ( 𝜑 → ( 𝑇 𝑃 𝑈 ) : ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ⟶ ( Base ‘ 𝐺 ) )
30 27 eleq2d ⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ 𝑥 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ) )
31 27 eleq2d ⊢ ( 𝜑 → ( 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ 𝑦 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ) )
32 30 31 anbi12d ⊢ ( 𝜑 → ( ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ↔ ( 𝑥 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ∧ 𝑦 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ) ) )
33 32 biimpar ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ∧ 𝑦 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ) ) → ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) )
34 1 2 3 4 5 6 7 8 9 pj1id ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ) → 𝑥 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ) )
35 34 adantrr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → 𝑥 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ) )
36 1 2 3 4 5 6 7 8 9 pj1id ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) → 𝑦 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) )
37 36 adantrl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → 𝑦 = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) )
38 35 37 oveq12d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( 𝑥 + 𝑦 ) = ( ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ) + ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) ) )
39 5 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
40 grpmnd ⊢ ( 𝐺 ∈ Grp → 𝐺 ∈ Mnd )
41 39 20 40 3syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → 𝐺 ∈ Mnd )
42 39 23 syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → 𝑇 ⊆ ( Base ‘ 𝐺 ) )
43 simpl ⊢ ( ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) → 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) )
44 ffvelrn ⊢ ( ( ( 𝑇 𝑃 𝑈 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑇 ∧ 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) ∈ 𝑇 )
45 22 43 44 syl2an ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) ∈ 𝑇 )
46 42 45 sseldd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐺 ) )
47 simpr ⊢ ( ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) → 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) )
48 ffvelrn ⊢ ( ( ( 𝑇 𝑃 𝑈 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑇 ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ∈ 𝑇 )
49 22 47 48 syl2an ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ∈ 𝑇 )
50 42 49 sseldd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) )
51 6 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
52 11 subgss ⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) )
53 51 52 syl ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → 𝑈 ⊆ ( Base ‘ 𝐺 ) )
54 1 2 3 4 5 6 7 8 9 pj2f ⊢ ( 𝜑 → ( 𝑈 𝑃 𝑇 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑈 )
55 ffvelrn ⊢ ( ( ( 𝑈 𝑃 𝑇 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑈 ∧ 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ∈ 𝑈 )
56 54 43 55 syl2an ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ∈ 𝑈 )
57 53 56 sseldd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ∈ ( Base ‘ 𝐺 ) )
58 ffvelrn ⊢ ( ( ( 𝑈 𝑃 𝑇 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑈 ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ∈ 𝑈 )
59 54 47 58 syl2an ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ∈ 𝑈 )
60 53 59 sseldd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ∈ ( Base ‘ 𝐺 ) )
61 8 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) )
62 61 49 sseldd ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ∈ ( 𝑍 ‘ 𝑈 ) )
63 1 4 cntzi ⊢ ( ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ∈ ( 𝑍 ‘ 𝑈 ) ∧ ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ∈ 𝑈 ) → ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ) = ( ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) )
64 62 56 63 syl2anc ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ) = ( ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) )
65 11 1 41 46 50 57 60 64 mnd4g ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) + ( ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) ) = ( ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ) + ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) ) )
66 38 65 eqtr4d ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( 𝑥 + 𝑦 ) = ( ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) + ( ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) ) )
67 7 adantr ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( 𝑇 ∩ 𝑈 ) = { 0 } )
68 1 subgcl ⊢ ( ( ( 𝑇 ⊕ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) → ( 𝑥 + 𝑦 ) ∈ ( 𝑇 ⊕ 𝑈 ) )
69 68 3expb ⊢ ( ( ( 𝑇 ⊕ 𝑈 ) ∈ ( SubGrp ‘ 𝐺 ) ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( 𝑥 + 𝑦 ) ∈ ( 𝑇 ⊕ 𝑈 ) )
70 17 69 sylan ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( 𝑥 + 𝑦 ) ∈ ( 𝑇 ⊕ 𝑈 ) )
71 1 subgcl ⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) ∈ 𝑇 ∧ ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ∈ 𝑇 ) → ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) ∈ 𝑇 )
72 39 45 49 71 syl3anc ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) ∈ 𝑇 )
73 1 subgcl ⊢ ( ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) ∈ 𝑈 ∧ ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ∈ 𝑈 ) → ( ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) ∈ 𝑈 )
74 51 56 59 73 syl3anc ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) ∈ 𝑈 )
75 1 2 3 4 39 51 67 61 9 70 72 74 pj1eq ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑥 + 𝑦 ) = ( ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) + ( ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) ) ↔ ( ( ( 𝑇 𝑃 𝑈 ) ‘ ( 𝑥 + 𝑦 ) ) = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) ∧ ( ( 𝑈 𝑃 𝑇 ) ‘ ( 𝑥 + 𝑦 ) ) = ( ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) ) ) )
76 66 75 mpbid ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( ( 𝑇 𝑃 𝑈 ) ‘ ( 𝑥 + 𝑦 ) ) = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) ∧ ( ( 𝑈 𝑃 𝑇 ) ‘ ( 𝑥 + 𝑦 ) ) = ( ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑥 ) + ( ( 𝑈 𝑃 𝑇 ) ‘ 𝑦 ) ) ) )
77 76 simpld ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ∧ 𝑦 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ ( 𝑥 + 𝑦 ) ) = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) )
78 33 77 syldan ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ∧ 𝑦 ∈ ( Base ‘ ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) ) ) ) → ( ( 𝑇 𝑃 𝑈 ) ‘ ( 𝑥 + 𝑦 ) ) = ( ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑥 ) + ( ( 𝑇 𝑃 𝑈 ) ‘ 𝑦 ) ) )
79 10 11 15 1 19 21 29 78 isghmd ⊢ ( 𝜑 → ( 𝑇 𝑃 𝑈 ) ∈ ( ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) GrpHom 𝐺 ) )