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

Theorem unitpropd 19137
Description: The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
Hypotheses
Ref Expression
rngidpropd.1 (𝜑𝐵 = (Base‘𝐾))
rngidpropd.2 (𝜑𝐵 = (Base‘𝐿))
rngidpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
Assertion
Ref Expression
unitpropd (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem unitpropd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 rngidpropd.1 . . . . . . 7 (𝜑𝐵 = (Base‘𝐾))
2 rngidpropd.2 . . . . . . 7 (𝜑𝐵 = (Base‘𝐿))
3 rngidpropd.3 . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
41, 2, 3rngidpropd 19135 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
54breq2d 4974 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐾) ↔ 𝑧(∥r𝐾)(1r𝐿)))
64breq2d 4974 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐾) ↔ 𝑧(∥r‘(oppr𝐾))(1r𝐿)))
75, 6anbi12d 630 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿))))
81, 2, 3dvdsrpropd 19136 . . . . . 6 (𝜑 → (∥r𝐾) = (∥r𝐿))
98breqd 4973 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐿) ↔ 𝑧(∥r𝐿)(1r𝐿)))
10 eqid 2795 . . . . . . . . 9 (oppr𝐾) = (oppr𝐾)
11 eqid 2795 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
1210, 11opprbas 19069 . . . . . . . 8 (Base‘𝐾) = (Base‘(oppr𝐾))
131, 12syl6eq 2847 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐾)))
14 eqid 2795 . . . . . . . . 9 (oppr𝐿) = (oppr𝐿)
15 eqid 2795 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1614, 15opprbas 19069 . . . . . . . 8 (Base‘𝐿) = (Base‘(oppr𝐿))
172, 16syl6eq 2847 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐿)))
183ancom2s 646 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
19 eqid 2795 . . . . . . . . 9 (.r𝐾) = (.r𝐾)
20 eqid 2795 . . . . . . . . 9 (.r‘(oppr𝐾)) = (.r‘(oppr𝐾))
2111, 19, 10, 20opprmul 19066 . . . . . . . 8 (𝑦(.r‘(oppr𝐾))𝑥) = (𝑥(.r𝐾)𝑦)
22 eqid 2795 . . . . . . . . 9 (.r𝐿) = (.r𝐿)
23 eqid 2795 . . . . . . . . 9 (.r‘(oppr𝐿)) = (.r‘(oppr𝐿))
2415, 22, 14, 23opprmul 19066 . . . . . . . 8 (𝑦(.r‘(oppr𝐿))𝑥) = (𝑥(.r𝐿)𝑦)
2518, 21, 243eqtr4g 2856 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑦(.r‘(oppr𝐾))𝑥) = (𝑦(.r‘(oppr𝐿))𝑥))
2613, 17, 25dvdsrpropd 19136 . . . . . 6 (𝜑 → (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐿)))
2726breqd 4973 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐿) ↔ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
289, 27anbi12d 630 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
297, 28bitrd 280 . . 3 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
30 eqid 2795 . . . 4 (Unit‘𝐾) = (Unit‘𝐾)
31 eqid 2795 . . . 4 (1r𝐾) = (1r𝐾)
32 eqid 2795 . . . 4 (∥r𝐾) = (∥r𝐾)
33 eqid 2795 . . . 4 (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐾))
3430, 31, 32, 10, 33isunit 19097 . . 3 (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)))
35 eqid 2795 . . . 4 (Unit‘𝐿) = (Unit‘𝐿)
36 eqid 2795 . . . 4 (1r𝐿) = (1r𝐿)
37 eqid 2795 . . . 4 (∥r𝐿) = (∥r𝐿)
38 eqid 2795 . . . 4 (∥r‘(oppr𝐿)) = (∥r‘(oppr𝐿))
3935, 36, 37, 14, 38isunit 19097 . . 3 (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
4029, 34, 393bitr4g 315 . 2 (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿)))
4140eqrdv 2793 1 (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081   class class class wbr 4962  cfv 6225  (class class class)co 7016  Basecbs 16312  .rcmulr 16395  1rcur 18941  opprcoppr 19062  rcdsr 19078  Unitcui 19079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-tpos 7743  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-plusg 16407  df-mulr 16408  df-0g 16544  df-mgp 18930  df-ur 18942  df-oppr 19063  df-dvdsr 19081  df-unit 19082
This theorem is referenced by:  invrpropd  19138  drngprop  19203  drngpropd  19219  sradrng  30592
  Copyright terms: Public domain W3C validator