Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rhmsscmap2 Structured version   Visualization version   GIF version

Theorem rhmsscmap2 42498
Description: The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
Hypotheses
Ref Expression
rhmsscmap.u (𝜑𝑈𝑉)
rhmsscmap.r (𝜑𝑅 = (Ring ∩ 𝑈))
Assertion
Ref Expression
rhmsscmap2 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Distinct variable group:   𝑥,𝑅,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rhmsscmap2
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3753 . . 3 𝑅𝑅
21a1i 11 . 2 (𝜑𝑅𝑅)
3 eqid 2748 . . . . . . 7 (Base‘𝑎) = (Base‘𝑎)
4 eqid 2748 . . . . . . 7 (Base‘𝑏) = (Base‘𝑏)
53, 4rhmf 18899 . . . . . 6 ( ∈ (𝑎 RingHom 𝑏) → :(Base‘𝑎)⟶(Base‘𝑏))
6 simpr 479 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → :(Base‘𝑎)⟶(Base‘𝑏))
7 fvex 6350 . . . . . . . . . 10 (Base‘𝑏) ∈ V
8 fvex 6350 . . . . . . . . . 10 (Base‘𝑎) ∈ V
97, 8pm3.2i 470 . . . . . . . . 9 ((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V)
10 elmapg 8024 . . . . . . . . 9 (((Base‘𝑏) ∈ V ∧ (Base‘𝑎) ∈ V) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
119, 10mp1i 13 . . . . . . . 8 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ( ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ↔ :(Base‘𝑎)⟶(Base‘𝑏)))
126, 11mpbird 247 . . . . . . 7 (((𝜑 ∧ (𝑎𝑅𝑏𝑅)) ∧ :(Base‘𝑎)⟶(Base‘𝑏)) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
1312ex 449 . . . . . 6 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (:(Base‘𝑎)⟶(Base‘𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
145, 13syl5 34 . . . . 5 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → ( ∈ (𝑎 RingHom 𝑏) → ∈ ((Base‘𝑏) ↑𝑚 (Base‘𝑎))))
1514ssrdv 3738 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎 RingHom 𝑏) ⊆ ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
16 ovres 6953 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
1716adantl 473 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) = (𝑎 RingHom 𝑏))
18 eqidd 2749 . . . . . 6 ((𝑎𝑅𝑏𝑅) → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
19 fveq2 6340 . . . . . . . 8 (𝑦 = 𝑏 → (Base‘𝑦) = (Base‘𝑏))
20 fveq2 6340 . . . . . . . 8 (𝑥 = 𝑎 → (Base‘𝑥) = (Base‘𝑎))
2119, 20oveqan12rd 6821 . . . . . . 7 ((𝑥 = 𝑎𝑦 = 𝑏) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2221adantl 473 . . . . . 6 (((𝑎𝑅𝑏𝑅) ∧ (𝑥 = 𝑎𝑦 = 𝑏)) → ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
23 simpl 474 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑎𝑅)
24 simpr 479 . . . . . 6 ((𝑎𝑅𝑏𝑅) → 𝑏𝑅)
25 ovexd 6831 . . . . . 6 ((𝑎𝑅𝑏𝑅) → ((Base‘𝑏) ↑𝑚 (Base‘𝑎)) ∈ V)
2618, 22, 23, 24, 25ovmpt2d 6941 . . . . 5 ((𝑎𝑅𝑏𝑅) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2726adantl 473 . . . 4 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏) = ((Base‘𝑏) ↑𝑚 (Base‘𝑎)))
2815, 17, 273sstr4d 3777 . . 3 ((𝜑 ∧ (𝑎𝑅𝑏𝑅)) → (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
2928ralrimivva 3097 . 2 (𝜑 → ∀𝑎𝑅𝑏𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))
30 rhmfn 42397 . . . . 5 RingHom Fn (Ring × Ring)
3130a1i 11 . . . 4 (𝜑 → RingHom Fn (Ring × Ring))
32 rhmsscmap.r . . . . . 6 (𝜑𝑅 = (Ring ∩ 𝑈))
33 inss1 3964 . . . . . 6 (Ring ∩ 𝑈) ⊆ Ring
3432, 33syl6eqss 3784 . . . . 5 (𝜑𝑅 ⊆ Ring)
35 xpss12 5269 . . . . 5 ((𝑅 ⊆ Ring ∧ 𝑅 ⊆ Ring) → (𝑅 × 𝑅) ⊆ (Ring × Ring))
3634, 34, 35syl2anc 696 . . . 4 (𝜑 → (𝑅 × 𝑅) ⊆ (Ring × Ring))
37 fnssres 6153 . . . 4 (( RingHom Fn (Ring × Ring) ∧ (𝑅 × 𝑅) ⊆ (Ring × Ring)) → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
3831, 36, 37syl2anc 696 . . 3 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) Fn (𝑅 × 𝑅))
39 eqid 2748 . . . . 5 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) = (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))
40 ovex 6829 . . . . 5 ((Base‘𝑦) ↑𝑚 (Base‘𝑥)) ∈ V
4139, 40fnmpt2i 7395 . . . 4 (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑅 × 𝑅)
4241a1i 11 . . 3 (𝜑 → (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) Fn (𝑅 × 𝑅))
43 incom 3936 . . . . 5 (Ring ∩ 𝑈) = (𝑈 ∩ Ring)
44 rhmsscmap.u . . . . . 6 (𝜑𝑈𝑉)
45 inex1g 4941 . . . . . 6 (𝑈𝑉 → (𝑈 ∩ Ring) ∈ V)
4644, 45syl 17 . . . . 5 (𝜑 → (𝑈 ∩ Ring) ∈ V)
4743, 46syl5eqel 2831 . . . 4 (𝜑 → (Ring ∩ 𝑈) ∈ V)
4832, 47eqeltrd 2827 . . 3 (𝜑𝑅 ∈ V)
4938, 42, 48isssc 16652 . 2 (𝜑 → (( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))) ↔ (𝑅𝑅 ∧ ∀𝑎𝑅𝑏𝑅 (𝑎( RingHom ↾ (𝑅 × 𝑅))𝑏) ⊆ (𝑎(𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))𝑏))))
502, 29, 49mpbir2and 995 1 (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1620  wcel 2127  wral 3038  Vcvv 3328  cin 3702  wss 3703   class class class wbr 4792   × cxp 5252  cres 5256   Fn wfn 6032  wf 6033  cfv 6037  (class class class)co 6801  cmpt2 6803  𝑚 cmap 8011  Basecbs 16030  cat cssc 16639  Ringcrg 18718   RingHom crh 18885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-cnex 10155  ax-resscn 10156  ax-1cn 10157  ax-icn 10158  ax-addcl 10159  ax-addrcl 10160  ax-mulcl 10161  ax-mulrcl 10162  ax-mulcom 10163  ax-addass 10164  ax-mulass 10165  ax-distr 10166  ax-i2m1 10167  ax-1ne0 10168  ax-1rid 10169  ax-rnegex 10170  ax-rrecex 10171  ax-cnre 10172  ax-pre-lttri 10173  ax-pre-lttrn 10174  ax-pre-ltadd 10175  ax-pre-mulgt0 10176
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-nel 3024  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-riota 6762  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-om 7219  df-1st 7321  df-2nd 7322  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-er 7899  df-map 8013  df-ixp 8063  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10239  df-mnf 10240  df-xr 10241  df-ltxr 10242  df-le 10243  df-sub 10431  df-neg 10432  df-nn 11184  df-2 11242  df-ndx 16033  df-slot 16034  df-base 16036  df-sets 16037  df-plusg 16127  df-0g 16275  df-ssc 16642  df-mhm 17507  df-ghm 17830  df-mgp 18661  df-ur 18673  df-ring 18720  df-rnghom 18888
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator