Theorem rhmply1vr1 22331

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 5861	. . 3 ⊢ (𝑝 = 𝑋 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑋))
3		rhmply1vr1.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))
4		rhmrcl1 20427	. . . . 5 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
5	3, 4	syl 17	. . . 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 22160	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
10	5, 9	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	6	fvexi 6910	. . . . 5 ⊢ 𝑋 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
13	3, 12	coexd 7939	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
14	1, 2, 10, 13	fvmptd3 7027	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
15		eqid 2725	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2725	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	15, 16	rhmf 20436	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
18	3, 17	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
19		eqid 2725	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅)
20	15, 19	ringidcl 20214	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
21	5, 20	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
22		eqid 2725	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
23	15, 22	ring0cl 20215	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
24	5, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
25	21, 24	ifcld 4576	. . . . . . 7 ⊢ (𝜑 → if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
26	25	adantr 479	. . . . . 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 7141	. . . . 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 6912	. . . . . . 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 2725	. . . . . . . . . 10 ⊢ (1r‘𝑆) = (1r‘𝑆)
30	19, 29	rhm1 20440	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
31	3, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
32		rhmghm 20435	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
33		eqid 2725	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
34	22, 33	ghmid 19185	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 GrpHom 𝑆) → (𝐻‘(0g‘𝑅)) = (0g‘𝑆))
35	3, 32, 34	3syl 18	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(0g‘𝑅)) = (0g‘𝑆))
36	31, 35	ifeq12d 4551	. . . . . . 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 2777	. . . . . 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 5251	. . . . 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 2765	. . . 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 2725	. . . . . 6 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅)
41		eqid 2725	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
42		1oex 8497	. . . . . . 7 ⊢ 1o ∈ V
43	42	a1i 11	. . . . . 6 ⊢ (𝜑 → 1o ∈ V)
44		0lt1o 8525	. . . . . . 7 ⊢ ∅ ∈ 1o
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 1o)
46	40, 41, 22, 19, 43, 5, 45	mvrval 21944	. . . . 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 5865	. . . 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 2725	. . . . 5 ⊢ (1o mVar 𝑆) = (1o mVar 𝑆)
49		rhmrcl2 20428	. . . . . 6 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
50	3, 49	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ Ring)
51	48, 41, 33, 29, 43, 50, 45	mvrval 21944	. . . 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 2775	. . 3 ⊢ (𝜑 → (𝐻 ∘ ((1o mVar 𝑅)‘∅)) = ((1o mVar 𝑆)‘∅))
53	6	vr1val 22134	. . . 4 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)
54	53	coeq2i 5863	. . 3 ⊢ (𝐻 ∘ 𝑋) = (𝐻 ∘ ((1o mVar 𝑅)‘∅))
55		rhmply1vr1.y	. . . 4 ⊢ 𝑌 = (var1‘𝑆)
56	55	vr1val 22134	. . 3 ⊢ 𝑌 = ((1o mVar 𝑆)‘∅)
57	52, 54, 56	3eqtr4g 2790	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝑋) = 𝑌)
58	14, 57	eqtrd 2765	1 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {crab 3418  Vcvv 3461  ∅c0 4322  ifcif 4530   ↦ cmpt 5232  ^◡ccnv 5677   “ cima 5681   ∘ ccom 5682  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  1_oc1o 8480   ↑_m cmap 8845  Fincfn 8964  0cc0 11140  1c1 11141  ℕcn 12245  ℕ₀cn0 12505  Basecbs 17183  0_gc0g 17424   GrpHom cghm 19175  1_rcur 20133  Ringcrg 20185   RingHom crh 20420   mVar cmvr 21855  var₁cv1 22118  Poly₁cpl1 22119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-of 7685  df-om 7872  df-1st 7994  df-2nd 7995  df-supp 8166  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-fsupp 9388  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-7 12313  df-8 12314  df-9 12315  df-n0 12506  df-z 12592  df-dec 12711  df-uz 12856  df-fz 13520  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-tset 17255  df-ple 17256  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-mhm 18743  df-grp 18901  df-ghm 19176  df-mgp 20087  df-ur 20134  df-ring 20187  df-rhm 20423  df-psr 21859  df-mvr 21860  df-mpl 21861  df-opsr 21863  df-psr1 22122  df-vr1 22123  df-ply1 22124

This theorem is referenced by:  rhmply1mon  22333