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Theorem rhmply1vsca 22408
Description: Apply a ring homomorphism between two univariate polynomial algebras to a scaled polynomial. (Contributed by SN, 20-May-2025.)
Hypotheses
Ref Expression
rhmply1vsca.p 𝑃 = (Poly1𝑅)
rhmply1vsca.q 𝑄 = (Poly1𝑆)
rhmply1vsca.b 𝐵 = (Base‘𝑃)
rhmply1vsca.k 𝐾 = (Base‘𝑅)
rhmply1vsca.f 𝐹 = (𝑝𝐵 ↦ (𝐻𝑝))
rhmply1vsca.t · = ( ·𝑠𝑃)
rhmply1vsca.u = ( ·𝑠𝑄)
rhmply1vsca.h (𝜑𝐻 ∈ (𝑅 RingHom 𝑆))
rhmply1vsca.c (𝜑𝐶𝐾)
rhmply1vsca.x (𝜑𝑋𝐵)
Assertion
Ref Expression
rhmply1vsca (𝜑 → (𝐹‘(𝐶 · 𝑋)) = ((𝐻𝐶) (𝐹𝑋)))
Distinct variable groups:   𝐶,𝑝   𝑋,𝑝   𝐻,𝑝   𝐵,𝑝   · ,𝑝
Allowed substitution hints:   𝜑(𝑝)   𝑃(𝑝)   𝑄(𝑝)   𝑅(𝑝)   𝑆(𝑝)   (𝑝)   𝐹(𝑝)   𝐾(𝑝)

Proof of Theorem rhmply1vsca
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rhmply1vsca.c . . . . . . . 8 (𝜑𝐶𝐾)
2 fconst6g 6798 . . . . . . . 8 (𝐶𝐾 → ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶}):{ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin}⟶𝐾)
31, 2syl 17 . . . . . . 7 (𝜑 → ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶}):{ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin}⟶𝐾)
4 psr1baslem 22202 . . . . . . . 8 (ℕ0m 1o) = { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin}
54feq2i 6729 . . . . . . 7 (({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶}):(ℕ0m 1o)⟶𝐾 ↔ ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶}):{ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin}⟶𝐾)
63, 5sylibr 234 . . . . . 6 (𝜑 → ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶}):(ℕ0m 1o)⟶𝐾)
7 rhmply1vsca.x . . . . . . 7 (𝜑𝑋𝐵)
8 rhmply1vsca.p . . . . . . . 8 𝑃 = (Poly1𝑅)
9 rhmply1vsca.b . . . . . . . 8 𝐵 = (Base‘𝑃)
10 rhmply1vsca.k . . . . . . . 8 𝐾 = (Base‘𝑅)
118, 9, 10ply1basf 22220 . . . . . . 7 (𝑋𝐵𝑋:(ℕ0m 1o)⟶𝐾)
127, 11syl 17 . . . . . 6 (𝜑𝑋:(ℕ0m 1o)⟶𝐾)
13 rhmply1vsca.h . . . . . . . 8 (𝜑𝐻 ∈ (𝑅 RingHom 𝑆))
14 eqid 2735 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
1510, 14rhmf 20502 . . . . . . . 8 (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻:𝐾⟶(Base‘𝑆))
1613, 15syl 17 . . . . . . 7 (𝜑𝐻:𝐾⟶(Base‘𝑆))
1716ffnd 6738 . . . . . 6 (𝜑𝐻 Fn 𝐾)
18 ovexd 7466 . . . . . 6 (𝜑 → (ℕ0m 1o) ∈ V)
19 rhmrcl1 20493 . . . . . . . . 9 (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
2013, 19syl 17 . . . . . . . 8 (𝜑𝑅 ∈ Ring)
21 eqid 2735 . . . . . . . . 9 (.r𝑅) = (.r𝑅)
2210, 21ringcl 20268 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑎𝐾𝑏𝐾) → (𝑎(.r𝑅)𝑏) ∈ 𝐾)
2320, 22syl3an1 1162 . . . . . . 7 ((𝜑𝑎𝐾𝑏𝐾) → (𝑎(.r𝑅)𝑏) ∈ 𝐾)
24233expb 1119 . . . . . 6 ((𝜑 ∧ (𝑎𝐾𝑏𝐾)) → (𝑎(.r𝑅)𝑏) ∈ 𝐾)
25 eqid 2735 . . . . . . . . 9 (.r𝑆) = (.r𝑆)
2610, 21, 25rhmmul 20503 . . . . . . . 8 ((𝐻 ∈ (𝑅 RingHom 𝑆) ∧ 𝑎𝐾𝑏𝐾) → (𝐻‘(𝑎(.r𝑅)𝑏)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑏)))
2713, 26syl3an1 1162 . . . . . . 7 ((𝜑𝑎𝐾𝑏𝐾) → (𝐻‘(𝑎(.r𝑅)𝑏)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑏)))
28273expb 1119 . . . . . 6 ((𝜑 ∧ (𝑎𝐾𝑏𝐾)) → (𝐻‘(𝑎(.r𝑅)𝑏)) = ((𝐻𝑎)(.r𝑆)(𝐻𝑏)))
296, 12, 17, 18, 24, 28coof 7721 . . . . 5 (𝜑 → (𝐻 ∘ (({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r𝑅)𝑋)) = ((𝐻 ∘ ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶})) ∘f (.r𝑆)(𝐻𝑋)))
30 fcoconst 7154 . . . . . . 7 ((𝐻 Fn 𝐾𝐶𝐾) → (𝐻 ∘ ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶})) = ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {(𝐻𝐶)}))
3117, 1, 30syl2anc 584 . . . . . 6 (𝜑 → (𝐻 ∘ ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶})) = ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {(𝐻𝐶)}))
3231oveq1d 7446 . . . . 5 (𝜑 → ((𝐻 ∘ ({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶})) ∘f (.r𝑆)(𝐻𝑋)) = (({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {(𝐻𝐶)}) ∘f (.r𝑆)(𝐻𝑋)))
3329, 32eqtrd 2775 . . . 4 (𝜑 → (𝐻 ∘ (({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r𝑅)𝑋)) = (({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {(𝐻𝐶)}) ∘f (.r𝑆)(𝐻𝑋)))
34 eqid 2735 . . . . . 6 (1o mPoly 𝑅) = (1o mPoly 𝑅)
35 eqid 2735 . . . . . 6 ( ·𝑠 ‘(1o mPoly 𝑅)) = ( ·𝑠 ‘(1o mPoly 𝑅))
368, 9ply1bas 22212 . . . . . 6 𝐵 = (Base‘(1o mPoly 𝑅))
37 eqid 2735 . . . . . 6 { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin}
3834, 35, 10, 36, 21, 37, 1, 7mplvsca 22053 . . . . 5 (𝜑 → (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋) = (({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r𝑅)𝑋))
3938coeq2d 5876 . . . 4 (𝜑 → (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)) = (𝐻 ∘ (({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {𝐶}) ∘f (.r𝑅)𝑋)))
40 eqid 2735 . . . . 5 (1o mPoly 𝑆) = (1o mPoly 𝑆)
41 eqid 2735 . . . . 5 ( ·𝑠 ‘(1o mPoly 𝑆)) = ( ·𝑠 ‘(1o mPoly 𝑆))
42 rhmply1vsca.q . . . . . 6 𝑄 = (Poly1𝑆)
43 eqid 2735 . . . . . 6 (Base‘𝑄) = (Base‘𝑄)
4442, 43ply1bas 22212 . . . . 5 (Base‘𝑄) = (Base‘(1o mPoly 𝑆))
4516, 1ffvelcdmd 7105 . . . . 5 (𝜑 → (𝐻𝐶) ∈ (Base‘𝑆))
46 rhmghm 20501 . . . . . . 7 (𝐻 ∈ (𝑅 RingHom 𝑆) → 𝐻 ∈ (𝑅 GrpHom 𝑆))
47 ghmmhm 19257 . . . . . . 7 (𝐻 ∈ (𝑅 GrpHom 𝑆) → 𝐻 ∈ (𝑅 MndHom 𝑆))
4813, 46, 473syl 18 . . . . . 6 (𝜑𝐻 ∈ (𝑅 MndHom 𝑆))
498, 42, 9, 43, 48, 7mhmcoply1 22405 . . . . 5 (𝜑 → (𝐻𝑋) ∈ (Base‘𝑄))
5040, 41, 14, 44, 25, 37, 45, 49mplvsca 22053 . . . 4 (𝜑 → ((𝐻𝐶)( ·𝑠 ‘(1o mPoly 𝑆))(𝐻𝑋)) = (({ ∈ (ℕ0m 1o) ∣ ( “ ℕ) ∈ Fin} × {(𝐻𝐶)}) ∘f (.r𝑆)(𝐻𝑋)))
5133, 39, 503eqtr4d 2785 . . 3 (𝜑 → (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)) = ((𝐻𝐶)( ·𝑠 ‘(1o mPoly 𝑆))(𝐻𝑋)))
52 rhmply1vsca.t . . . . . 6 · = ( ·𝑠𝑃)
538, 34, 52ply1vsca 22242 . . . . 5 · = ( ·𝑠 ‘(1o mPoly 𝑅))
5453oveqi 7444 . . . 4 (𝐶 · 𝑋) = (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋)
5554coeq2i 5874 . . 3 (𝐻 ∘ (𝐶 · 𝑋)) = (𝐻 ∘ (𝐶( ·𝑠 ‘(1o mPoly 𝑅))𝑋))
56 rhmply1vsca.u . . . . 5 = ( ·𝑠𝑄)
5742, 40, 56ply1vsca 22242 . . . 4 = ( ·𝑠 ‘(1o mPoly 𝑆))
5857oveqi 7444 . . 3 ((𝐻𝐶) (𝐻𝑋)) = ((𝐻𝐶)( ·𝑠 ‘(1o mPoly 𝑆))(𝐻𝑋))
5951, 55, 583eqtr4g 2800 . 2 (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) = ((𝐻𝐶) (𝐻𝑋)))
60 rhmply1vsca.f . . 3 𝐹 = (𝑝𝐵 ↦ (𝐻𝑝))
61 coeq2 5872 . . 3 (𝑝 = (𝐶 · 𝑋) → (𝐻𝑝) = (𝐻 ∘ (𝐶 · 𝑋)))
628, 9, 10, 52, 20, 1, 7ply1vscl 22404 . . 3 (𝜑 → (𝐶 · 𝑋) ∈ 𝐵)
6313, 62coexd 7954 . . 3 (𝜑 → (𝐻 ∘ (𝐶 · 𝑋)) ∈ V)
6460, 61, 62, 63fvmptd3 7039 . 2 (𝜑 → (𝐹‘(𝐶 · 𝑋)) = (𝐻 ∘ (𝐶 · 𝑋)))
65 coeq2 5872 . . . 4 (𝑝 = 𝑋 → (𝐻𝑝) = (𝐻𝑋))
6613, 7coexd 7954 . . . 4 (𝜑 → (𝐻𝑋) ∈ V)
6760, 65, 7, 66fvmptd3 7039 . . 3 (𝜑 → (𝐹𝑋) = (𝐻𝑋))
6867oveq2d 7447 . 2 (𝜑 → ((𝐻𝐶) (𝐹𝑋)) = ((𝐻𝐶) (𝐻𝑋)))
6959, 64, 683eqtr4d 2785 1 (𝜑 → (𝐹‘(𝐶 · 𝑋)) = ((𝐻𝐶) (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  {csn 4631  cmpt 5231   × cxp 5687  ccnv 5688  cima 5692  ccom 5693   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  1oc1o 8498  m cmap 8865  Fincfn 8984  cn 12264  0cn0 12524  Basecbs 17245  .rcmulr 17299   ·𝑠 cvsca 17302   MndHom cmhm 18807   GrpHom cghm 19243  Ringcrg 20251   RingHom crh 20486   mPoly cmpl 21944  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-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  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-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-prds 17494  df-pws 17496  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-minusg 18968  df-sbg 18969  df-subg 19154  df-ghm 19244  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-ring 20253  df-rhm 20489  df-lmod 20877  df-lss 20948  df-psr 21947  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-ply1 22199
This theorem is referenced by:  rhmply1mon  22409
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