Theorem funcringcsetcALTV2lem7

Description: Lemma 7 for funcringcsetcALTV2 . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

	Ref	Expression
Hypotheses	funcringcsetcALTV2.r	⊢ 𝑅 = ( RingCat ‘ 𝑈 )
	funcringcsetcALTV2.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
	funcringcsetcALTV2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	funcringcsetcALTV2.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
	funcringcsetcALTV2.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
	funcringcsetcALTV2.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
	funcringcsetcALTV2.g	⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
Assertion	funcringcsetcALTV2lem7	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 𝐺 𝑋 ) ‘ ( ( Id ‘ 𝑅 ) ‘ 𝑋 ) ) = ( ( Id ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		funcringcsetcALTV2.r	⊢ 𝑅 = ( RingCat ‘ 𝑈 )
2		funcringcsetcALTV2.s	⊢ 𝑆 = ( SetCat ‘ 𝑈 )
3		funcringcsetcALTV2.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
4		funcringcsetcALTV2.c	⊢ 𝐶 = ( Base ‘ 𝑆 )
5		funcringcsetcALTV2.u	⊢ ( 𝜑 → 𝑈 ∈ WUni )
6		funcringcsetcALTV2.f	⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( Base ‘ 𝑥 ) ) )
7		funcringcsetcALTV2.g	⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ ( I ↾ ( 𝑥 RingHom 𝑦 ) ) ) )
8	1 2 3 4 5 6 7	funcringcsetcALTV2lem5	⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) ) → ( 𝑋 𝐺 𝑋 ) = ( I ↾ ( 𝑋 RingHom 𝑋 ) ) )
9	8	anabsan2	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 𝐺 𝑋 ) = ( I ↾ ( 𝑋 RingHom 𝑋 ) ) )
10		eqid	⊢ ( Id ‘ 𝑅 ) = ( Id ‘ 𝑅 )
11	5	adantr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑈 ∈ WUni )
12		simpr	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
13		eqid	⊢ ( Base ‘ 𝑋 ) = ( Base ‘ 𝑋 )
14	1 3 10 11 12 13	ringcid	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( Id ‘ 𝑅 ) ‘ 𝑋 ) = ( I ↾ ( Base ‘ 𝑋 ) ) )
15	9 14	fveq12d	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 𝐺 𝑋 ) ‘ ( ( Id ‘ 𝑅 ) ‘ 𝑋 ) ) = ( ( I ↾ ( 𝑋 RingHom 𝑋 ) ) ‘ ( I ↾ ( Base ‘ 𝑋 ) ) ) )
16	1 3 5	ringcbas	⊢ ( 𝜑 → 𝐵 = ( 𝑈 ∩ Ring ) )
17	16	eleq2d	⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 ↔ 𝑋 ∈ ( 𝑈 ∩ Ring ) ) )
18		elin	⊢ ( 𝑋 ∈ ( 𝑈 ∩ Ring ) ↔ ( 𝑋 ∈ 𝑈 ∧ 𝑋 ∈ Ring ) )
19	18	simprbi	⊢ ( 𝑋 ∈ ( 𝑈 ∩ Ring ) → 𝑋 ∈ Ring )
20	17 19	syl6bi	⊢ ( 𝜑 → ( 𝑋 ∈ 𝐵 → 𝑋 ∈ Ring ) )
21	20	imp	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ Ring )
22	13	idrhm	⊢ ( 𝑋 ∈ Ring → ( I ↾ ( Base ‘ 𝑋 ) ) ∈ ( 𝑋 RingHom 𝑋 ) )
23		fvresi	⊢ ( ( I ↾ ( Base ‘ 𝑋 ) ) ∈ ( 𝑋 RingHom 𝑋 ) → ( ( I ↾ ( 𝑋 RingHom 𝑋 ) ) ‘ ( I ↾ ( Base ‘ 𝑋 ) ) ) = ( I ↾ ( Base ‘ 𝑋 ) ) )
24	21 22 23	3syl	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( I ↾ ( 𝑋 RingHom 𝑋 ) ) ‘ ( I ↾ ( Base ‘ 𝑋 ) ) ) = ( I ↾ ( Base ‘ 𝑋 ) ) )
25	1 2 3 4 5 6	funcringcsetcALTV2lem1	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) = ( Base ‘ 𝑋 ) )
26	25	fveq2d	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( Id ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑋 ) ) = ( ( Id ‘ 𝑆 ) ‘ ( Base ‘ 𝑋 ) ) )
27		eqid	⊢ ( Id ‘ 𝑆 ) = ( Id ‘ 𝑆 )
28	1 3 5	ringcbasbas	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( Base ‘ 𝑋 ) ∈ 𝑈 )
29	2 27 11 28	setcid	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( Id ‘ 𝑆 ) ‘ ( Base ‘ 𝑋 ) ) = ( I ↾ ( Base ‘ 𝑋 ) ) )
30	26 29	eqtr2d	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( I ↾ ( Base ‘ 𝑋 ) ) = ( ( Id ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑋 ) ) )
31	15 24 30	3eqtrd	⊢ ( ( 𝜑 ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑋 𝐺 𝑋 ) ‘ ( ( Id ‘ 𝑅 ) ‘ 𝑋 ) ) = ( ( Id ‘ 𝑆 ) ‘ ( 𝐹 ‘ 𝑋 ) ) )