Theorem rhmply1vr1 22509

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 5842	. . 3 ⊢ (𝑝 = 𝑋 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑋))
3		rhmply1vr1.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))
4		rhmrcl1 20554	. . . . 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 22342	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
10	5, 9	syl 18	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	6	fvexi 6893	. . . . 5 ⊢ 𝑋 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
13	3, 12	coexd 7924	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
14	1, 2, 10, 13	fvmptd3 7011	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
15		eqid 2769	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2769	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	15, 16	rhmf 20562	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
18	3, 17	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
19		eqid 2769	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅)
20	15, 19	ringidcl 20344	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
21	5, 20	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
22		eqid 2769	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
23	15, 22	ring0cl 20346	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
24	5, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
25	21, 24	ifcld 4536	. . . . . . 7 ⊢ (𝜑 → if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
26	25	adantr 485	. . . . . 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 7126	. . . . 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 6895	. . . . . . 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 2769	. . . . . . . . . 10 ⊢ (1r‘𝑆) = (1r‘𝑆)
30	19, 29	rhm1 20567	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
31	3, 30	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
32		rhmghm 20561	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
33		eqid 2769	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
34	22, 33	ghmid 19288	. . . . . . . . 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 4511	. . . . . . 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 2816	. . . . . 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 5206	. . . . 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 2804	. . . 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 2769	. . . . . 6 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅)
41		eqid 2769	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
42		1oex 8459	. . . . . . 7 ⊢ 1o ∈ V
43	42	a1i 11	. . . . . 6 ⊢ (𝜑 → 1o ∈ V)
44		0lt1o 8485	. . . . . . 7 ⊢ ∅ ∈ 1o
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 1o)
46	40, 41, 22, 19, 43, 5, 45	mvrval 22096	. . . . 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 5846	. . . 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 2769	. . . . 5 ⊢ (1o mVar 𝑆) = (1o mVar 𝑆)
49		rhmrcl2 20555	. . . . . 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 22096	. . . 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 2814	. . 3 ⊢ (𝜑 → (𝐻 ∘ ((1o mVar 𝑅)‘∅)) = ((1o mVar 𝑆)‘∅))
53	6	vr1val 22317	. . . 4 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)
54	53	coeq2i 5844	. . 3 ⊢ (𝐻 ∘ 𝑋) = (𝐻 ∘ ((1o mVar 𝑅)‘∅))
55		rhmply1vr1.y	. . . 4 ⊢ 𝑌 = (var1‘𝑆)
56	55	vr1val 22317	. . 3 ⊢ 𝑌 = ((1o mVar 𝑆)‘∅)
57	52, 54, 56	3eqtr4g 2829	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝑋) = 𝑌)
58	14, 57	eqtrd 2804	1 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {crab 3423  Vcvv 3463  ∅c0 4294  ifcif 4489   ↦ cmpt 5193  ^◡ccnv 5658   “ cima 5662   ∘ ccom 5663  ⟶wf 6530  ‘cfv 6534  (class class class)co 7408  1_oc1o 8442   ↑_m cmap 8820  Fincfn 8939  0cc0 11096  1c1 11097  ℕcn 12229  ℕ₀cn0 12500  Basecbs 17265  0_gc0g 17488   GrpHom cghm 19279  1_rcur 20259  Ringcrg 20311   RingHom crh 20547   mVar cmvr 22020  var₁cv1 22301  Poly₁cpl1 22302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730  ax-cnex 11152  ax-resscn 11153  ax-1cn 11154  ax-icn 11155  ax-addcl 11156  ax-addrcl 11157  ax-mulcl 11158  ax-mulrcl 11159  ax-mulcom 11160  ax-addass 11161  ax-mulass 11162  ax-distr 11163  ax-i2m1 11164  ax-1ne0 11165  ax-1rid 11166  ax-rnegex 11167  ax-rrecex 11168  ax-cnre 11169  ax-pre-lttri 11170  ax-pre-lttrn 11171  ax-pre-ltadd 11172  ax-pre-mulgt0 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6300  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6490  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-of 7672  df-om 7859  df-1st 7982  df-2nd 7983  df-supp 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9318  df-pnf 11241  df-mnf 11242  df-xr 11243  df-ltxr 11244  df-le 11245  df-sub 11439  df-neg 11440  df-nn 12230  df-2 12299  df-3 12300  df-4 12301  df-5 12302  df-6 12303  df-7 12304  df-8 12305  df-9 12306  df-n0 12501  df-z 12588  df-dec 12708  df-uz 12859  df-fz 13532  df-struct 17203  df-sets 17220  df-slot 17238  df-ndx 17250  df-base 17266  df-ress 17287  df-plusg 17319  df-mulr 17320  df-sca 17322  df-vsca 17323  df-tset 17325  df-ple 17326  df-0g 17490  df-mgm 18694  df-sgrp 18773  df-mnd 18789  df-mhm 18837  df-grp 18999  df-ghm 19280  df-mgp 20213  df-ur 20260  df-ring 20313  df-rhm 20550  df-psr 22024  df-mvr 22025  df-mpl 22026  df-opsr 22028  df-psr1 22305  df-vr1 22306  df-ply1 22307

This theorem is referenced by:  rhmply1mon  22511  aks5lem3a  42841