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Theorem unitpropd 18520
 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 18518 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
54breq2d 4595 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐾) ↔ 𝑧(∥r𝐾)(1r𝐿)))
64breq2d 4595 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐾) ↔ 𝑧(∥r‘(oppr𝐾))(1r𝐿)))
75, 6anbi12d 743 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿))))
81, 2, 3dvdsrpropd 18519 . . . . . 6 (𝜑 → (∥r𝐾) = (∥r𝐿))
98breqd 4594 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐿) ↔ 𝑧(∥r𝐿)(1r𝐿)))
10 eqid 2610 . . . . . . . . 9 (oppr𝐾) = (oppr𝐾)
11 eqid 2610 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
1210, 11opprbas 18452 . . . . . . . 8 (Base‘𝐾) = (Base‘(oppr𝐾))
131, 12syl6eq 2660 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐾)))
14 eqid 2610 . . . . . . . . 9 (oppr𝐿) = (oppr𝐿)
15 eqid 2610 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1614, 15opprbas 18452 . . . . . . . 8 (Base‘𝐿) = (Base‘(oppr𝐿))
172, 16syl6eq 2660 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐿)))
183ancom2s 840 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
19 eqid 2610 . . . . . . . . 9 (.r𝐾) = (.r𝐾)
20 eqid 2610 . . . . . . . . 9 (.r‘(oppr𝐾)) = (.r‘(oppr𝐾))
2111, 19, 10, 20opprmul 18449 . . . . . . . 8 (𝑦(.r‘(oppr𝐾))𝑥) = (𝑥(.r𝐾)𝑦)
22 eqid 2610 . . . . . . . . 9 (.r𝐿) = (.r𝐿)
23 eqid 2610 . . . . . . . . 9 (.r‘(oppr𝐿)) = (.r‘(oppr𝐿))
2415, 22, 14, 23opprmul 18449 . . . . . . . 8 (𝑦(.r‘(oppr𝐿))𝑥) = (𝑥(.r𝐿)𝑦)
2518, 21, 243eqtr4g 2669 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑦(.r‘(oppr𝐾))𝑥) = (𝑦(.r‘(oppr𝐿))𝑥))
2613, 17, 25dvdsrpropd 18519 . . . . . 6 (𝜑 → (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐿)))
2726breqd 4594 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐿) ↔ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
289, 27anbi12d 743 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
297, 28bitrd 267 . . 3 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
30 eqid 2610 . . . 4 (Unit‘𝐾) = (Unit‘𝐾)
31 eqid 2610 . . . 4 (1r𝐾) = (1r𝐾)
32 eqid 2610 . . . 4 (∥r𝐾) = (∥r𝐾)
33 eqid 2610 . . . 4 (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐾))
3430, 31, 32, 10, 33isunit 18480 . . 3 (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)))
35 eqid 2610 . . . 4 (Unit‘𝐿) = (Unit‘𝐿)
36 eqid 2610 . . . 4 (1r𝐿) = (1r𝐿)
37 eqid 2610 . . . 4 (∥r𝐿) = (∥r𝐿)
38 eqid 2610 . . . 4 (∥r‘(oppr𝐿)) = (∥r‘(oppr𝐿))
3935, 36, 37, 14, 38isunit 18480 . . 3 (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
4029, 34, 393bitr4g 302 . 2 (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿)))
4140eqrdv 2608 1 (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  1rcur 18324  opprcoppr 18445  ∥rcdsr 18461  Unitcui 18462 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgp 18313  df-ur 18325  df-oppr 18446  df-dvdsr 18464  df-unit 18465 This theorem is referenced by:  invrpropd  18521  drngprop  18581  drngpropd  18597
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