Theorem erngdvlem2-rN

Description: Lemma for eringring . (Contributed by NM, 6-Aug-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ernggrp.h-r	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ernggrp.d-r	⊢ 𝐷 = ( ( EDRingR ‘ 𝐾 ) ‘ 𝑊 )
	ernggrplem.b-r	⊢ 𝐵 = ( Base ‘ 𝐾 )
	ernggrplem.t-r	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	ernggrplem.e-r	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	ernggrplem.p-r	⊢ 𝑃 = ( 𝑎 ∈ 𝐸 , 𝑏 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑎 ‘ 𝑓 ) ∘ ( 𝑏 ‘ 𝑓 ) ) ) )
	ernggrplem.o-r	⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
	ernggrplem.i-r	⊢ 𝐼 = ( 𝑎 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ◡ ( 𝑎 ‘ 𝑓 ) ) )
Assertion	erngdvlem2-rN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Abel )

Proof

Step	Hyp	Ref	Expression
1		ernggrp.h-r	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		ernggrp.d-r	⊢ 𝐷 = ( ( EDRingR ‘ 𝐾 ) ‘ 𝑊 )
3		ernggrplem.b-r	⊢ 𝐵 = ( Base ‘ 𝐾 )
4		ernggrplem.t-r	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		ernggrplem.e-r	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
6		ernggrplem.p-r	⊢ 𝑃 = ( 𝑎 ∈ 𝐸 , 𝑏 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑎 ‘ 𝑓 ) ∘ ( 𝑏 ‘ 𝑓 ) ) ) )
7		ernggrplem.o-r	⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
8		ernggrplem.i-r	⊢ 𝐼 = ( 𝑎 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ◡ ( 𝑎 ‘ 𝑓 ) ) )
9		eqid	⊢ ( Base ‘ 𝐷 ) = ( Base ‘ 𝐷 )
10	1 4 5 2 9	erngbase-rN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( Base ‘ 𝐷 ) = 𝐸 )
11	10	eqcomd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐸 = ( Base ‘ 𝐷 ) )
12		eqid	⊢ ( +g ‘ 𝐷 ) = ( +g ‘ 𝐷 )
13	1 4 5 2 12	erngfplus-rN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +g ‘ 𝐷 ) = ( 𝑎 ∈ 𝐸 , 𝑏 ∈ 𝐸 ↦ ( 𝑓 ∈ 𝑇 ↦ ( ( 𝑎 ‘ 𝑓 ) ∘ ( 𝑏 ‘ 𝑓 ) ) ) ) )
14	6 13	eqtr4id	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑃 = ( +g ‘ 𝐷 ) )
15	1 2 3 4 5 6 7 8	erngdvlem1-rN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Grp )
16	1 4 5 6	tendoplcom	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ∧ 𝑡 ∈ 𝐸 ) → ( 𝑠 𝑃 𝑡 ) = ( 𝑡 𝑃 𝑠 ) )
17	11 14 15 16	isabld	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 ∈ Abel )