MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rhmply1vr1 Structured version   Visualization version   GIF version

Theorem rhmply1vr1 22391
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 = (Poly1𝑅)
rhmply1vr1.q 𝑄 = (Poly1𝑆)
rhmply1vr1.b 𝐵 = (Base‘𝑃)
rhmply1vr1.f 𝐹 = (𝑝𝐵 ↦ (𝐻𝑝))
rhmply1vr1.x 𝑋 = (var1𝑅)
rhmply1vr1.y 𝑌 = (var1𝑆)
rhmply1vr1.h (𝜑𝐻 ∈ (𝑅 RingHom 𝑆))
Assertion
Ref Expression
rhmply1vr1 (𝜑 → (𝐹𝑋) = 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 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rhmply1vr1.f . . 3 𝐹 = (𝑝𝐵 ↦ (𝐻𝑝))
2 coeq2 5869 . . 3 (𝑝 = 𝑋 → (𝐻𝑝) = (𝐻𝑋))
3 rhmply1vr1.h . . . . 5 (𝜑𝐻 ∈ (𝑅 RingHom 𝑆))
4 rhmrcl1 20476 . . . . 5 (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
53, 4syl 17 . . . 4 (𝜑𝑅 ∈ Ring)
6 rhmply1vr1.x . . . . 5 𝑋 = (var1𝑅)
7 rhmply1vr1.p . . . . 5 𝑃 = (Poly1𝑅)
8 rhmply1vr1.b . . . . 5 𝐵 = (Base‘𝑃)
96, 7, 8vr1cl 22219 . . . 4 (𝑅 ∈ Ring → 𝑋𝐵)
105, 9syl 17 . . 3 (𝜑𝑋𝐵)
116fvexi 6920 . . . . 5 𝑋 ∈ V
1211a1i 11 . . . 4 (𝜑𝑋 ∈ V)
133, 12coexd 7953 . . 3 (𝜑 → (𝐻𝑋) ∈ V)
141, 2, 10, 13fvmptd3 7039 . 2 (𝜑 → (𝐹𝑋) = (𝐻𝑋))
15 eqid 2737 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
16 eqid 2737 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
1715, 16rhmf 20485 . . . . . . 7 (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:(Base‘𝑅)⟶(Base‘𝑆))
183, 17syl 17 . . . . . 6 (𝜑𝐻:(Base‘𝑅)⟶(Base‘𝑆))
19 eqid 2737 . . . . . . . . . 10 (1r𝑅) = (1r𝑅)
2015, 19ringidcl 20262 . . . . . . . . 9 (𝑅 ∈ Ring → (1r𝑅) ∈ (Base‘𝑅))
215, 20syl 17 . . . . . . . 8 (𝜑 → (1r𝑅) ∈ (Base‘𝑅))
22 eqid 2737 . . . . . . . . . 10 (0g𝑅) = (0g𝑅)
2315, 22ring0cl 20264 . . . . . . . . 9 (𝑅 ∈ Ring → (0g𝑅) ∈ (Base‘𝑅))
245, 23syl 17 . . . . . . . 8 (𝜑 → (0g𝑅) ∈ (Base‘𝑅))
2521, 24ifcld 4572 . . . . . . 7 (𝜑 → if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅)) ∈ (Base‘𝑅))
2625adantr 480 . . . . . 6 ((𝜑𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin}) → if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅)) ∈ (Base‘𝑅))
2718, 26cofmpt 7152 . . . . 5 (𝜑 → (𝐻 ∘ (𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅)))) = (𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐻‘if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅)))))
28 fvif 6922 . . . . . . 7 (𝐻‘if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅))) = if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (𝐻‘(1r𝑅)), (𝐻‘(0g𝑅)))
29 eqid 2737 . . . . . . . . . 10 (1r𝑆) = (1r𝑆)
3019, 29rhm1 20489 . . . . . . . . 9 (𝐻 ∈ (𝑅 RingHom 𝑆) → (𝐻‘(1r𝑅)) = (1r𝑆))
313, 30syl 17 . . . . . . . 8 (𝜑 → (𝐻‘(1r𝑅)) = (1r𝑆))
32 rhmghm 20484 . . . . . . . . 9 (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
33 eqid 2737 . . . . . . . . . 10 (0g𝑆) = (0g𝑆)
3422, 33ghmid 19240 . . . . . . . . 9 (𝐻 ∈ (𝑅 GrpHom 𝑆) → (𝐻‘(0g𝑅)) = (0g𝑆))
353, 32, 343syl 18 . . . . . . . 8 (𝜑 → (𝐻‘(0g𝑅)) = (0g𝑆))
3631, 35ifeq12d 4547 . . . . . . 7 (𝜑 → if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (𝐻‘(1r𝑅)), (𝐻‘(0g𝑅))) = if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑆), (0g𝑆)))
3728, 36eqtrid 2789 . . . . . 6 (𝜑 → (𝐻‘if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅))) = if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑆), (0g𝑆)))
3837mpteq2dv 5244 . . . . 5 (𝜑 → (𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} ↦ (𝐻‘if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅)))) = (𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑆), (0g𝑆))))
3927, 38eqtrd 2777 . . . 4 (𝜑 → (𝐻 ∘ (𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅)))) = (𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑆), (0g𝑆))))
40 eqid 2737 . . . . . 6 (1o mVar 𝑅) = (1o mVar 𝑅)
41 eqid 2737 . . . . . 6 { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin}
42 1oex 8516 . . . . . . 7 1o ∈ V
4342a1i 11 . . . . . 6 (𝜑 → 1o ∈ V)
44 0lt1o 8542 . . . . . . 7 ∅ ∈ 1o
4544a1i 11 . . . . . 6 (𝜑 → ∅ ∈ 1o)
4640, 41, 22, 19, 43, 5, 45mvrval 22002 . . . . 5 (𝜑 → ((1o mVar 𝑅)‘∅) = (𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅))))
4746coeq2d 5873 . . . 4 (𝜑 → (𝐻 ∘ ((1o mVar 𝑅)‘∅)) = (𝐻 ∘ (𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑅), (0g𝑅)))))
48 eqid 2737 . . . . 5 (1o mVar 𝑆) = (1o mVar 𝑆)
49 rhmrcl2 20477 . . . . . 6 (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
503, 49syl 17 . . . . 5 (𝜑𝑆 ∈ Ring)
5148, 41, 33, 29, 43, 50, 45mvrval 22002 . . . 4 (𝜑 → ((1o mVar 𝑆)‘∅) = (𝑓 ∈ { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} ↦ if(𝑓 = (𝑦 ∈ 1o ↦ if(𝑦 = ∅, 1, 0)), (1r𝑆), (0g𝑆))))
5239, 47, 513eqtr4d 2787 . . 3 (𝜑 → (𝐻 ∘ ((1o mVar 𝑅)‘∅)) = ((1o mVar 𝑆)‘∅))
536vr1val 22193 . . . 4 𝑋 = ((1o mVar 𝑅)‘∅)
5453coeq2i 5871 . . 3 (𝐻𝑋) = (𝐻 ∘ ((1o mVar 𝑅)‘∅))
55 rhmply1vr1.y . . . 4 𝑌 = (var1𝑆)
5655vr1val 22193 . . 3 𝑌 = ((1o mVar 𝑆)‘∅)
5752, 54, 563eqtr4g 2802 . 2 (𝜑 → (𝐻𝑋) = 𝑌)
5814, 57eqtrd 2777 1 (𝜑 → (𝐹𝑋) = 𝑌)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  {crab 3436  Vcvv 3480  c0 4333  ifcif 4525  cmpt 5225  ccnv 5684  cima 5688  ccom 5689  wf 6557  cfv 6561  (class class class)co 7431  1oc1o 8499  m cmap 8866  Fincfn 8985  0cc0 11155  1c1 11156  cn 12266  0cn0 12526  Basecbs 17247  0gc0g 17484   GrpHom cghm 19230  1rcur 20178  Ringcrg 20230   RingHom crh 20469   mVar cmvr 21925  var1cv1 22177  Poly1cpl1 22178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-tset 17316  df-ple 17317  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-ghm 19231  df-mgp 20138  df-ur 20179  df-ring 20232  df-rhm 20472  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-psr1 22181  df-vr1 22182  df-ply1 22183
This theorem is referenced by:  rhmply1mon  22393  aks5lem3a  42190
  Copyright terms: Public domain W3C validator