This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem pj1ghm2

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 pj1ghm2 ( 𝜑 → ( 𝑇 𝑃 𝑈 ) ∈ ( ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) GrpHom ( 𝐺 ↾s 𝑇 ) ) )

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 1 2 3 4 5 6 7 8 9 pj1ghm ⊢ ( 𝜑 → ( 𝑇 𝑃 𝑈 ) ∈ ( ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) GrpHom 𝐺 ) )
11 1 2 3 4 5 6 7 8 9 pj1f ⊢ ( 𝜑 → ( 𝑇 𝑃 𝑈 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑇 )
12 11 frnd ⊢ ( 𝜑 → ran ( 𝑇 𝑃 𝑈 ) ⊆ 𝑇 )
13 eqid ⊢ ( 𝐺 ↾s 𝑇 ) = ( 𝐺 ↾s 𝑇 )
14 13 resghm2b ⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ ran ( 𝑇 𝑃 𝑈 ) ⊆ 𝑇 ) → ( ( 𝑇 𝑃 𝑈 ) ∈ ( ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) GrpHom 𝐺 ) ↔ ( 𝑇 𝑃 𝑈 ) ∈ ( ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) GrpHom ( 𝐺 ↾s 𝑇 ) ) ) )
15 5 12 14 syl2anc ⊢ ( 𝜑 → ( ( 𝑇 𝑃 𝑈 ) ∈ ( ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) GrpHom 𝐺 ) ↔ ( 𝑇 𝑃 𝑈 ) ∈ ( ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) GrpHom ( 𝐺 ↾s 𝑇 ) ) ) )
16 10 15 mpbid ⊢ ( 𝜑 → ( 𝑇 𝑃 𝑈 ) ∈ ( ( 𝐺 ↾s ( 𝑇 ⊕ 𝑈 ) ) GrpHom ( 𝐺 ↾s 𝑇 ) ) )