Theorem rhmply1vr1 22610

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 20619	. . . . 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 22443	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
10	5, 9	syl 18	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	6	fvexi 6896	. . . . 5 ⊢ 𝑋 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
13	3, 12	coexd 7931	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
14	1, 2, 10, 13	fvmptd3 7014	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
15		eqid 2762	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2762	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	15, 16	rhmf 20627	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
18	3, 17	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
19		eqid 2762	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅)
20	15, 19	ringidcl 20407	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
21	5, 20	syl 18	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
22		eqid 2762	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
23	15, 22	ring0cl 20409	. . . . . . . . 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 7129	. . . . 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 6898	. . . . . . 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 2762	. . . . . . . . . 10 ⊢ (1r‘𝑆) = (1r‘𝑆)
30	19, 29	rhm1 20636	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
31	3, 30	syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
32		rhmghm 20626	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
33		eqid 2762	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
34	22, 33	ghmid 19350	. . . . . . . . 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 2809	. . . . . 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 5203	. . . . 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 2797	. . . 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 2762	. . . . . 6 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅)
41		eqid 2762	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
42		1oex 8468	. . . . . . 7 ⊢ 1o ∈ V
43	42	a1i 11	. . . . . 6 ⊢ (𝜑 → 1o ∈ V)
44		0lt1o 8494	. . . . . . 7 ⊢ ∅ ∈ 1o
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 1o)
46	40, 41, 22, 19, 43, 5, 45	mvrval 22197	. . . . 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 2762	. . . . 5 ⊢ (1o mVar 𝑆) = (1o mVar 𝑆)
49		rhmrcl2 20620	. . . . . 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 22197	. . . 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 2807	. . 3 ⊢ (𝜑 → (𝐻 ∘ ((1o mVar 𝑅)‘∅)) = ((1o mVar 𝑆)‘∅))
53	6	vr1val 22418	. . . 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 22418	. . 3 ⊢ 𝑌 = ((1o mVar 𝑆)‘∅)
57	52, 54, 56	3eqtr4g 2822	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝑋) = 𝑌)
58	14, 57	eqtrd 2797	1 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3414  Vcvv 3453  ∅c0 4282  ifcif 4485   ↦ cmpt 5190  ^◡ccnv 5658   “ cima 5662   ∘ ccom 5663  ⟶wf 6533  ‘cfv 6537  (class class class)co 7416  1_oc1o 8451   ↑_m cmap 8829  Fincfn 8955  0cc0 11127  1c1 11128  ℕcn 12260  ℕ₀cn0 12531  Basecbs 17305  0_gc0g 17528   GrpHom cghm 19341  1_rcur 20321  Ringcrg 20373   RingHom crh 20611   mVar cmvr 22121  var₁cv1 22402  Poly₁cpl1 22403

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7681  df-om 7866  df-1st 7989  df-2nd 7990  df-supp 8162  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-fsupp 9335  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-z 12619  df-dec 12740  df-uz 12891  df-fz 13564  df-struct 17243  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-mulr 17360  df-sca 17362  df-vsca 17363  df-tset 17365  df-ple 17366  df-0g 17530  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-mhm 18892  df-grp 19061  df-ghm 19342  df-mgp 20275  df-ur 20322  df-ring 20375  df-rhm 20614  df-psr 22125  df-mvr 22126  df-mpl 22127  df-opsr 22129  df-psr1 22406  df-vr1 22407  df-ply1 22408

This theorem is used by:  rhmply1mon  22612  aks5lem3a  43042