Theorem rhmply1vr1 22729

Description: A ring homomorphism between two univariate polynomial algebras sends one variable to the other. (Contributed by SN, 20-May-2025.)

Hypotheses

Ref	Expression
rhmply1vr1.p	⊢ 𝑃 = (Poly1‘𝑅)
rhmply1vr1.q	⊢ 𝑄 = (Poly1‘𝑆)
rhmply1vr1.b	⊢ 𝐵 = (Base‘𝑃)
rhmply1vr1.f	⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
rhmply1vr1.x	⊢ 𝑋 = (var1‘𝑅)
rhmply1vr1.y	⊢ 𝑌 = (var1‘𝑆)
rhmply1vr1.h	⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))

Assertion

Ref	Expression
rhmply1vr1	⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

Distinct variable groups:   𝑋,𝑝   𝐻,𝑝   𝐵,𝑝

Allowed substitution hints:   𝜑(𝑝)   𝑃(𝑝)   𝑄(𝑝)   𝑅(𝑝)   𝑆(𝑝)   𝐹(𝑝)   𝑌(𝑝)

Proof of Theorem rhmply1vr1

Dummy variables 𝑓 ℎ 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rhmply1vr1.f	. . 3 ⊢ 𝐹 = (𝑝 ∈ 𝐵 ↦ (𝐻 ∘ 𝑝))
2		coeq2 5841	. . 3 ⊢ (𝑝 = 𝑋 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑋))
3		rhmply1vr1.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))
4		rhmrcl1 20734	. . . . 5 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
5	3, 4	syl 18	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
6		rhmply1vr1.x	. . . . 5 ⊢ 𝑋 = (var1‘𝑅)
7		rhmply1vr1.p	. . . . 5 ⊢ 𝑃 = (Poly1‘𝑅)
8		rhmply1vr1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑃)
9	6, 7, 8	vr1cl 22562	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
10	5, 9	syl 18	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	6	fvexi 6919	. . . . 5 ⊢ 𝑋 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
13	3, 12	coexd 7964	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
14	1, 2, 10, 13	fvmptd3 7037	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
15		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	15, 16	rhmf 20742	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
18	3, 17	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
19		eqid 2763	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅)
20	15, 19	ringidcl 20521	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
21	5, 20	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
22		eqid 2763	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
23	15, 22	ring0cl 20523	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
24	5, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
25	21, 24	ifcld 4532	. . . . . . 7 ⊢ (𝜑 → if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
26	25	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}) → if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
27	18, 26	cofmpt 7154	. . . . 5 ⊢ (𝜑 → (𝐻 ∘ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)))) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐻‘if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)))))
28		fvif 6921	. . . . . . 7 ⊢ (𝐻‘if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅))) = if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (𝐻‘(1r‘𝑅)), (𝐻‘(0g‘𝑅)))
29		eqid 2763	. . . . . . . . . 10 ⊢ (1r‘𝑆) = (1r‘𝑆)
30	19, 29	rhm1 20751	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
31	3, 30	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
32		rhmghm 20741	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
33		eqid 2763	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
34	22, 33	ghmid 19462	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 GrpHom 𝑆) → (𝐻‘(0g‘𝑅)) = (0g‘𝑆))
35	3, 32, 34	3syl 19	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(0g‘𝑅)) = (0g‘𝑆))
36	31, 35	ifeq12d 4507	. . . . . . 7 ⊢ (𝜑 → if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (𝐻‘(1r‘𝑅)), (𝐻‘(0g‘𝑅))) = if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑆), (0g‘𝑆)))
37	28, 36	eqtrid 2810	. . . . . 6 ⊢ (𝜑 → (𝐻‘if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅))) = if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑆), (0g‘𝑆)))
38	37	mpteq2dv 5202	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ (𝐻‘if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)))) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑆), (0g‘𝑆))))
39	27, 38	eqtrd 2798	. . . 4 ⊢ (𝜑 → (𝐻 ∘ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)))) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑆), (0g‘𝑆))))
40		eqid 2763	. . . . . 6 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅)
41		eqid 2763	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
42		1oex 8511	. . . . . . 7 ⊢ 1o ∈ V
43	42	a1i 11	. . . . . 6 ⊢ (𝜑 → 1o ∈ V)
44		0lt1o 8537	. . . . . . 7 ⊢ ∅ ∈ 1o
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 1o)
46	40, 41, 22, 19, 43, 5, 45	mvrval 22316	. . . . 5 ⊢ (𝜑 → ((1o mVar 𝑅)‘∅) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅))))
47	46	coeq2d 5845	. . . 4 ⊢ (𝜑 → (𝐻 ∘ ((1o mVar 𝑅)‘∅)) = (𝐻 ∘ (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)))))
48		eqid 2763	. . . . 5 ⊢ (1o mVar 𝑆) = (1o mVar 𝑆)
49		rhmrcl2 20735	. . . . . 6 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
50	3, 49	syl 18	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
51	48, 41, 33, 29, 43, 50, 45	mvrval 22316	. . . 4 ⊢ (𝜑 → ((1o mVar 𝑆)‘∅) = (𝑓 ∈ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑆), (0g‘𝑆))))
52	39, 47, 51	3eqtr4d 2808	. . 3 ⊢ (𝜑 → (𝐻 ∘ ((1o mVar 𝑅)‘∅)) = ((1o mVar 𝑆)‘∅))
53	6	vr1val 22537	. . . 4 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)
54	53	coeq2i 5843	. . 3 ⊢ (𝐻 ∘ 𝑋) = (𝐻 ∘ ((1o mVar 𝑅)‘∅))
55		rhmply1vr1.y	. . . 4 ⊢ 𝑌 = (var1‘𝑆)
56	55	vr1val 22537	. . 3 ⊢ 𝑌 = ((1o mVar 𝑆)‘∅)
57	52, 54, 56	3eqtr4g 2823	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝑋) = 𝑌)
58	14, 57	eqtrd 2798	1 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454  ∅c0 4281  ifcif 4485   ↦ cmpt 5189  ^◡ccnv 5654   “ cima 5658   ∘ ccom 5659  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  1_oc1o 8494   ↑_m cmap 8872  Fincfn 8998  0cc0 11224  1c1 11225  ℕcn 12359  ℕ₀cn0 12630  Basecbs 17413  0_gc0g 17636   GrpHom cghm 19453  1_rcur 20434  Ringcrg 20486   RingHom crh 20726   mVar cmvr 22240  var₁cv1 22521  Poly₁cpl1 22522

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-of 7714  df-om 7899  df-1st 8022  df-2nd 8023  df-supp 8202  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-1o 8501  df-er 8742  df-map 8874  df-en 8999  df-dom 9000  df-sdom 9001  df-fin 9002  df-fsupp 9379  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-struct 17351  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-ress 17435  df-plusg 17467  df-mulr 17468  df-sca 17470  df-vsca 17471  df-tset 17473  df-ple 17474  df-0g 17638  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-mhm 19004  df-grp 19173  df-ghm 19454  df-mgp 20388  df-ur 20435  df-ring 20488  df-rhm 20729  df-psr 22244  df-mvr 22245  df-mpl 22246  df-opsr 22248  df-psr1 22525  df-vr1 22526  df-ply1 22527

This theorem is used by:  rhmply1mon  22731  aks5lem3a  43430