Theorem rhmply1vr1 22407

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 5872	. . 3 ⊢ (𝑝 = 𝑋 → (𝐻 ∘ 𝑝) = (𝐻 ∘ 𝑋))
3		rhmply1vr1.h	. . . . 5 ⊢ (𝜑 → 𝐻 ∈ (𝑅 RingHom 𝑆))
4		rhmrcl1 20493	. . . . 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 22235	. . . 4 ⊢ (𝑅 ∈ Ring → 𝑋 ∈ 𝐵)
10	5, 9	syl 17	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	6	fvexi 6921	. . . . 5 ⊢ 𝑋 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
13	3, 12	coexd 7954	. . 3 ⊢ (𝜑 → (𝐻 ∘ 𝑋) ∈ V)
14	1, 2, 10, 13	fvmptd3 7039	. 2 ⊢ (𝜑 → (𝐹‘𝑋) = (𝐻 ∘ 𝑋))
15		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
16		eqid 2735	. . . . . . . 8 ⊢ (Base‘𝑆) = (Base‘𝑆)
17	15, 16	rhmf 20502	. . . . . . 7 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
18	3, 17	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
19		eqid 2735	. . . . . . . . . 10 ⊢ (1r‘𝑅) = (1r‘𝑅)
20	15, 19	ringidcl 20280	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ (Base‘𝑅))
21	5, 20	syl 17	. . . . . . . 8 ⊢ (𝜑 → (1r‘𝑅) ∈ (Base‘𝑅))
22		eqid 2735	. . . . . . . . . 10 ⊢ (0g‘𝑅) = (0g‘𝑅)
23	15, 22	ring0cl 20281	. . . . . . . . 9 ⊢ (𝑅 ∈ Ring → (0g‘𝑅) ∈ (Base‘𝑅))
24	5, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → (0g‘𝑅) ∈ (Base‘𝑅))
25	21, 24	ifcld 4577	. . . . . . 7 ⊢ (𝜑 → if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r‘𝑅), (0g‘𝑅)) ∈ (Base‘𝑅))
26	25	adantr 480	. . . . . 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 7152	. . . . 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 6923	. . . . . . 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 2735	. . . . . . . . . 10 ⊢ (1r‘𝑆) = (1r‘𝑆)
30	19, 29	rhm1 20506	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
31	3, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝐻‘(1r‘𝑅)) = (1r‘𝑆))
32		rhmghm 20501	. . . . . . . . 9 ⊢ (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
33		eqid 2735	. . . . . . . . . 10 ⊢ (0g‘𝑆) = (0g‘𝑆)
34	22, 33	ghmid 19253	. . . . . . . . 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 4552	. . . . . . 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 2787	. . . . . 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 5250	. . . . 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 2775	. . . 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 2735	. . . . . 6 ⊢ (1o mVar 𝑅) = (1o mVar 𝑅)
41		eqid 2735	. . . . . 6 ⊢ {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ0 ↑m 1o) ∣ (◡ℎ “ ℕ) ∈ Fin}
42		1oex 8515	. . . . . . 7 ⊢ 1o ∈ V
43	42	a1i 11	. . . . . 6 ⊢ (𝜑 → 1o ∈ V)
44		0lt1o 8541	. . . . . . 7 ⊢ ∅ ∈ 1o
45	44	a1i 11	. . . . . 6 ⊢ (𝜑 → ∅ ∈ 1o)
46	40, 41, 22, 19, 43, 5, 45	mvrval 22020	. . . . 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 5876	. . . 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 2735	. . . . 5 ⊢ (1o mVar 𝑆) = (1o mVar 𝑆)
49		rhmrcl2 20494	. . . . . 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 22020	. . . 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 2785	. . 3 ⊢ (𝜑 → (𝐻 ∘ ((1o mVar 𝑅)‘∅)) = ((1o mVar 𝑆)‘∅))
53	6	vr1val 22209	. . . 4 ⊢ 𝑋 = ((1o mVar 𝑅)‘∅)
54	53	coeq2i 5874	. . 3 ⊢ (𝐻 ∘ 𝑋) = (𝐻 ∘ ((1o mVar 𝑅)‘∅))
55		rhmply1vr1.y	. . . 4 ⊢ 𝑌 = (var1‘𝑆)
56	55	vr1val 22209	. . 3 ⊢ 𝑌 = ((1o mVar 𝑆)‘∅)
57	52, 54, 56	3eqtr4g 2800	. 2 ⊢ (𝜑 → (𝐻 ∘ 𝑋) = 𝑌)
58	14, 57	eqtrd 2775	1 ⊢ (𝜑 → (𝐹‘𝑋) = 𝑌)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  {crab 3433  Vcvv 3478  ∅c0 4339  ifcif 4531   ↦ cmpt 5231  ^◡ccnv 5688   “ cima 5692   ∘ ccom 5693  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  1_oc1o 8498   ↑_m cmap 8865  Fincfn 8984  0cc0 11153  1c1 11154  ℕcn 12264  ℕ₀cn0 12524  Basecbs 17245  0_gc0g 17486   GrpHom cghm 19243  1_rcur 20199  Ringcrg 20251   RingHom crh 20486   mVar cmvr 21943  var₁cv1 22193  Poly₁cpl1 22194

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-tset 17317  df-ple 17318  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-ghm 19244  df-mgp 20153  df-ur 20200  df-ring 20253  df-rhm 20489  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-vr1 22198  df-ply1 22199

This theorem is referenced by:  rhmply1mon  22409  aks5lem3a  42171