Theorem rhmply1mon

Description: Apply a ring homomorphism between two univariate polynomial algebras to a scaled monomial, as in ply1coe . (Contributed by SN, 20-May-2025)

	Ref	Expression
Hypotheses	rhmply1mon.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
	rhmply1mon.q	⊢ 𝑄 = ( Poly1 ‘ 𝑆 )
	rhmply1mon.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
	rhmply1mon.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
	rhmply1mon.f	⊢ 𝐹 = ( 𝑝 ∈ 𝐵 ↦ ( 𝐻 ∘ 𝑝 ) )
	rhmply1mon.x	⊢ 𝑋 = ( var1 ‘ 𝑅 )
	rhmply1mon.y	⊢ 𝑌 = ( var1 ‘ 𝑆 )
	rhmply1mon.t	⊢ · = ( ·𝑠 ‘ 𝑃 )
	rhmply1mon.u	⊢ ∙ = ( ·𝑠 ‘ 𝑄 )
	rhmply1mon.m	⊢ 𝑀 = ( mulGrp ‘ 𝑃 )
	rhmply1mon.n	⊢ 𝑁 = ( mulGrp ‘ 𝑄 )
	rhmply1mon.l	⊢ ↑ = ( .g ‘ 𝑀 )
	rhmply1mon.w	⊢ ∧ = ( .g ‘ 𝑁 )
	rhmply1mon.h	⊢ ( 𝜑 → 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) )
	rhmply1mon.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐾 )
	rhmply1mon.e	⊢ ( 𝜑 → 𝐸 ∈ ℕ0 )
Assertion	rhmply1mon	⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐶 · ( 𝐸 ↑ 𝑋 ) ) ) = ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐸 ∧ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		rhmply1mon.p	⊢ 𝑃 = ( Poly1 ‘ 𝑅 )
2		rhmply1mon.q	⊢ 𝑄 = ( Poly1 ‘ 𝑆 )
3		rhmply1mon.b	⊢ 𝐵 = ( Base ‘ 𝑃 )
4		rhmply1mon.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
5		rhmply1mon.f	⊢ 𝐹 = ( 𝑝 ∈ 𝐵 ↦ ( 𝐻 ∘ 𝑝 ) )
6		rhmply1mon.x	⊢ 𝑋 = ( var1 ‘ 𝑅 )
7		rhmply1mon.y	⊢ 𝑌 = ( var1 ‘ 𝑆 )
8		rhmply1mon.t	⊢ · = ( ·𝑠 ‘ 𝑃 )
9		rhmply1mon.u	⊢ ∙ = ( ·𝑠 ‘ 𝑄 )
10		rhmply1mon.m	⊢ 𝑀 = ( mulGrp ‘ 𝑃 )
11		rhmply1mon.n	⊢ 𝑁 = ( mulGrp ‘ 𝑄 )
12		rhmply1mon.l	⊢ ↑ = ( .g ‘ 𝑀 )
13		rhmply1mon.w	⊢ ∧ = ( .g ‘ 𝑁 )
14		rhmply1mon.h	⊢ ( 𝜑 → 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) )
15		rhmply1mon.c	⊢ ( 𝜑 → 𝐶 ∈ 𝐾 )
16		rhmply1mon.e	⊢ ( 𝜑 → 𝐸 ∈ ℕ0 )
17	10 3	mgpbas	⊢ 𝐵 = ( Base ‘ 𝑀 )
18		rhmrcl1	⊢ ( 𝐻 ∈ ( 𝑅 RingHom 𝑆 ) → 𝑅 ∈ Ring )
19	14 18	syl	⊢ ( 𝜑 → 𝑅 ∈ Ring )
20	1	ply1ring	⊢ ( 𝑅 ∈ Ring → 𝑃 ∈ Ring )
21	19 20	syl	⊢ ( 𝜑 → 𝑃 ∈ Ring )
22	10	ringmgp	⊢ ( 𝑃 ∈ Ring → 𝑀 ∈ Mnd )
23	21 22	syl	⊢ ( 𝜑 → 𝑀 ∈ Mnd )
24	6 1 3	vr1cl	⊢ ( 𝑅 ∈ Ring → 𝑋 ∈ 𝐵 )
25	19 24	syl	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
26	17 12 23 16 25	mulgnn0cld	⊢ ( 𝜑 → ( 𝐸 ↑ 𝑋 ) ∈ 𝐵 )
27	1 2 3 4 5 8 9 14 15 26	rhmply1vsca	⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐶 · ( 𝐸 ↑ 𝑋 ) ) ) = ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐹 ‘ ( 𝐸 ↑ 𝑋 ) ) ) )
28	1 2 3 5 14	rhmply1	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑃 RingHom 𝑄 ) )
29	10 11	rhmmhm	⊢ ( 𝐹 ∈ ( 𝑃 RingHom 𝑄 ) → 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) )
30	28 29	syl	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) )
31	17 12 13	mhmmulg	⊢ ( ( 𝐹 ∈ ( 𝑀 MndHom 𝑁 ) ∧ 𝐸 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ ( 𝐸 ↑ 𝑋 ) ) = ( 𝐸 ∧ ( 𝐹 ‘ 𝑋 ) ) )
32	30 16 25 31	syl3anc	⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐸 ↑ 𝑋 ) ) = ( 𝐸 ∧ ( 𝐹 ‘ 𝑋 ) ) )
33	1 2 3 5 6 7 14	rhmply1vr1	⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) = 𝑌 )
34	33	oveq2d	⊢ ( 𝜑 → ( 𝐸 ∧ ( 𝐹 ‘ 𝑋 ) ) = ( 𝐸 ∧ 𝑌 ) )
35	32 34	eqtrd	⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐸 ↑ 𝑋 ) ) = ( 𝐸 ∧ 𝑌 ) )
36	35	oveq2d	⊢ ( 𝜑 → ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐹 ‘ ( 𝐸 ↑ 𝑋 ) ) ) = ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐸 ∧ 𝑌 ) ) )
37	27 36	eqtrd	⊢ ( 𝜑 → ( 𝐹 ‘ ( 𝐶 · ( 𝐸 ↑ 𝑋 ) ) ) = ( ( 𝐻 ‘ 𝐶 ) ∙ ( 𝐸 ∧ 𝑌 ) ) )