Theorem evl1muld

Description: Polynomial evaluation builder for multiplication of polynomials. (Contributed by Mario Carneiro, 4-Jul-2015)

	Ref	Expression
Hypotheses	evl1addd.q	⊢ 𝑂 = ( eval1 ‘ 𝑅 )
	evl1addd.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	evl1addd.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
	evl1addd.u	⊢ 𝑈 = ( Base ‘ 𝑃 )
	evl1addd.1	⊢ ( 𝜑 → 𝑅 ∈ CRing )
	evl1addd.2	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	evl1addd.3	⊢ ( 𝜑 → ( 𝑀 ∈ 𝑈 ∧ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) = 𝑉 ) )
	evl1addd.4	⊢ ( 𝜑 → ( 𝑁 ∈ 𝑈 ∧ ( ( 𝑂 ‘ 𝑁 ) ‘ 𝑌 ) = 𝑊 ) )
	evl1muld.t	⊢ ∙ = ( .r ‘ 𝑃 )
	evl1muld.s	⊢ · = ( .r ‘ 𝑅 )
Assertion	evl1muld	⊢ ( 𝜑 → ( ( 𝑀 ∙ 𝑁 ) ∈ 𝑈 ∧ ( ( 𝑂 ‘ ( 𝑀 ∙ 𝑁 ) ) ‘ 𝑌 ) = ( 𝑉 · 𝑊 ) ) )

Proof

Step	Hyp	Ref	Expression
1		evl1addd.q	⊢ 𝑂 = ( eval1 ‘ 𝑅 )
2		evl1addd.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
3		evl1addd.b	⊢ 𝐵 = ( Base ‘ 𝑅 )
4		evl1addd.u	⊢ 𝑈 = ( Base ‘ 𝑃 )
5		evl1addd.1	⊢ ( 𝜑 → 𝑅 ∈ CRing )
6		evl1addd.2	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		evl1addd.3	⊢ ( 𝜑 → ( 𝑀 ∈ 𝑈 ∧ ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) = 𝑉 ) )
8		evl1addd.4	⊢ ( 𝜑 → ( 𝑁 ∈ 𝑈 ∧ ( ( 𝑂 ‘ 𝑁 ) ‘ 𝑌 ) = 𝑊 ) )
9		evl1muld.t	⊢ ∙ = ( .r ‘ 𝑃 )
10		evl1muld.s	⊢ · = ( .r ‘ 𝑅 )
11		eqid	⊢ ( 𝑅 ↑s 𝐵 ) = ( 𝑅 ↑s 𝐵 )
12	1 2 11 3	evl1rhm	⊢ ( 𝑅 ∈ CRing → 𝑂 ∈ ( 𝑃 RingHom ( 𝑅 ↑s 𝐵 ) ) )
13	5 12	syl	⊢ ( 𝜑 → 𝑂 ∈ ( 𝑃 RingHom ( 𝑅 ↑s 𝐵 ) ) )
14		rhmrcl1	⊢ ( 𝑂 ∈ ( 𝑃 RingHom ( 𝑅 ↑s 𝐵 ) ) → 𝑃 ∈ Ring )
15	13 14	syl	⊢ ( 𝜑 → 𝑃 ∈ Ring )
16	7	simpld	⊢ ( 𝜑 → 𝑀 ∈ 𝑈 )
17	8	simpld	⊢ ( 𝜑 → 𝑁 ∈ 𝑈 )
18	4 9	ringcl	⊢ ( ( 𝑃 ∈ Ring ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈 ) → ( 𝑀 ∙ 𝑁 ) ∈ 𝑈 )
19	15 16 17 18	syl3anc	⊢ ( 𝜑 → ( 𝑀 ∙ 𝑁 ) ∈ 𝑈 )
20		eqid	⊢ ( .r ‘ ( 𝑅 ↑s 𝐵 ) ) = ( .r ‘ ( 𝑅 ↑s 𝐵 ) )
21	4 9 20	rhmmul	⊢ ( ( 𝑂 ∈ ( 𝑃 RingHom ( 𝑅 ↑s 𝐵 ) ) ∧ 𝑀 ∈ 𝑈 ∧ 𝑁 ∈ 𝑈 ) → ( 𝑂 ‘ ( 𝑀 ∙ 𝑁 ) ) = ( ( 𝑂 ‘ 𝑀 ) ( .r ‘ ( 𝑅 ↑s 𝐵 ) ) ( 𝑂 ‘ 𝑁 ) ) )
22	13 16 17 21	syl3anc	⊢ ( 𝜑 → ( 𝑂 ‘ ( 𝑀 ∙ 𝑁 ) ) = ( ( 𝑂 ‘ 𝑀 ) ( .r ‘ ( 𝑅 ↑s 𝐵 ) ) ( 𝑂 ‘ 𝑁 ) ) )
23		eqid	⊢ ( Base ‘ ( 𝑅 ↑s 𝐵 ) ) = ( Base ‘ ( 𝑅 ↑s 𝐵 ) )
24	3	fvexi	⊢ 𝐵 ∈ V
25	24	a1i	⊢ ( 𝜑 → 𝐵 ∈ V )
26	4 23	rhmf	⊢ ( 𝑂 ∈ ( 𝑃 RingHom ( 𝑅 ↑s 𝐵 ) ) → 𝑂 : 𝑈 ⟶ ( Base ‘ ( 𝑅 ↑s 𝐵 ) ) )
27	13 26	syl	⊢ ( 𝜑 → 𝑂 : 𝑈 ⟶ ( Base ‘ ( 𝑅 ↑s 𝐵 ) ) )
28	27 16	ffvelrnd	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑀 ) ∈ ( Base ‘ ( 𝑅 ↑s 𝐵 ) ) )
29	27 17	ffvelrnd	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑁 ) ∈ ( Base ‘ ( 𝑅 ↑s 𝐵 ) ) )
30	11 23 5 25 28 29 10 20	pwsmulrval	⊢ ( 𝜑 → ( ( 𝑂 ‘ 𝑀 ) ( .r ‘ ( 𝑅 ↑s 𝐵 ) ) ( 𝑂 ‘ 𝑁 ) ) = ( ( 𝑂 ‘ 𝑀 ) ∘f · ( 𝑂 ‘ 𝑁 ) ) )
31	22 30	eqtrd	⊢ ( 𝜑 → ( 𝑂 ‘ ( 𝑀 ∙ 𝑁 ) ) = ( ( 𝑂 ‘ 𝑀 ) ∘f · ( 𝑂 ‘ 𝑁 ) ) )
32	31	fveq1d	⊢ ( 𝜑 → ( ( 𝑂 ‘ ( 𝑀 ∙ 𝑁 ) ) ‘ 𝑌 ) = ( ( ( 𝑂 ‘ 𝑀 ) ∘f · ( 𝑂 ‘ 𝑁 ) ) ‘ 𝑌 ) )
33	11 3 23 5 25 28	pwselbas	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑀 ) : 𝐵 ⟶ 𝐵 )
34	33	ffnd	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑀 ) Fn 𝐵 )
35	11 3 23 5 25 29	pwselbas	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑁 ) : 𝐵 ⟶ 𝐵 )
36	35	ffnd	⊢ ( 𝜑 → ( 𝑂 ‘ 𝑁 ) Fn 𝐵 )
37		fnfvof	⊢ ( ( ( ( 𝑂 ‘ 𝑀 ) Fn 𝐵 ∧ ( 𝑂 ‘ 𝑁 ) Fn 𝐵 ) ∧ ( 𝐵 ∈ V ∧ 𝑌 ∈ 𝐵 ) ) → ( ( ( 𝑂 ‘ 𝑀 ) ∘f · ( 𝑂 ‘ 𝑁 ) ) ‘ 𝑌 ) = ( ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) · ( ( 𝑂 ‘ 𝑁 ) ‘ 𝑌 ) ) )
38	34 36 25 6 37	syl22anc	⊢ ( 𝜑 → ( ( ( 𝑂 ‘ 𝑀 ) ∘f · ( 𝑂 ‘ 𝑁 ) ) ‘ 𝑌 ) = ( ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) · ( ( 𝑂 ‘ 𝑁 ) ‘ 𝑌 ) ) )
39	7	simprd	⊢ ( 𝜑 → ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) = 𝑉 )
40	8	simprd	⊢ ( 𝜑 → ( ( 𝑂 ‘ 𝑁 ) ‘ 𝑌 ) = 𝑊 )
41	39 40	oveq12d	⊢ ( 𝜑 → ( ( ( 𝑂 ‘ 𝑀 ) ‘ 𝑌 ) · ( ( 𝑂 ‘ 𝑁 ) ‘ 𝑌 ) ) = ( 𝑉 · 𝑊 ) )
42	32 38 41	3eqtrd	⊢ ( 𝜑 → ( ( 𝑂 ‘ ( 𝑀 ∙ 𝑁 ) ) ‘ 𝑌 ) = ( 𝑉 · 𝑊 ) )
43	19 42	jca	⊢ ( 𝜑 → ( ( 𝑀 ∙ 𝑁 ) ∈ 𝑈 ∧ ( ( 𝑂 ‘ ( 𝑀 ∙ 𝑁 ) ) ‘ 𝑌 ) = ( 𝑉 · 𝑊 ) ) )