Theorem pj2f

Description: The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015)

	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	pj2f	⊢ ( 𝜑 → ( 𝑈 𝑃 𝑇 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑈 )

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		incom	⊢ ( 𝑈 ∩ 𝑇 ) = ( 𝑇 ∩ 𝑈 )
11	10 7	syl5eq	⊢ ( 𝜑 → ( 𝑈 ∩ 𝑇 ) = { 0 } )
12	4 5 6 8	cntzrecd	⊢ ( 𝜑 → 𝑈 ⊆ ( 𝑍 ‘ 𝑇 ) )
13	1 2 3 4 6 5 11 12 9	pj1f	⊢ ( 𝜑 → ( 𝑈 𝑃 𝑇 ) : ( 𝑈 ⊕ 𝑇 ) ⟶ 𝑈 )
14	2 4	lsmcom2	⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) )
15	5 6 8 14	syl3anc	⊢ ( 𝜑 → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) )
16	15	feq2d	⊢ ( 𝜑 → ( ( 𝑈 𝑃 𝑇 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑈 ↔ ( 𝑈 𝑃 𝑇 ) : ( 𝑈 ⊕ 𝑇 ) ⟶ 𝑈 ) )
17	13 16	mpbird	⊢ ( 𝜑 → ( 𝑈 𝑃 𝑇 ) : ( 𝑇 ⊕ 𝑈 ) ⟶ 𝑈 )