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Theorem invrpropd 18521
Description: The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
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
invrpropd (𝜑 → (invr𝐾) = (invr𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem invrpropd
StepHypRef Expression
1 eqid 2610 . . . . 5 (Unit‘𝐾) = (Unit‘𝐾)
2 eqid 2610 . . . . 5 ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾))
31, 2unitgrpbas 18489 . . . 4 (Unit‘𝐾) = (Base‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))
43a1i 11 . . 3 (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
5 rngidpropd.1 . . . . 5 (𝜑𝐵 = (Base‘𝐾))
6 rngidpropd.2 . . . . 5 (𝜑𝐵 = (Base‘𝐿))
7 rngidpropd.3 . . . . 5 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
85, 6, 7unitpropd 18520 . . . 4 (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
9 eqid 2610 . . . . 5 (Unit‘𝐿) = (Unit‘𝐿)
10 eqid 2610 . . . . 5 ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿))
119, 10unitgrpbas 18489 . . . 4 (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))
128, 11syl6eq 2660 . . 3 (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
13 eqid 2610 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
1413, 1unitss 18483 . . . . . . . 8 (Unit‘𝐾) ⊆ (Base‘𝐾)
1514, 5syl5sseqr 3617 . . . . . . 7 (𝜑 → (Unit‘𝐾) ⊆ 𝐵)
1615sselda 3568 . . . . . 6 ((𝜑𝑥 ∈ (Unit‘𝐾)) → 𝑥𝐵)
1715sselda 3568 . . . . . 6 ((𝜑𝑦 ∈ (Unit‘𝐾)) → 𝑦𝐵)
1816, 17anim12dan 878 . . . . 5 ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥𝐵𝑦𝐵))
1918, 7syldan 486 . . . 4 ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
20 fvex 6113 . . . . . 6 (Unit‘𝐾) ∈ V
21 eqid 2610 . . . . . . . 8 (mulGrp‘𝐾) = (mulGrp‘𝐾)
22 eqid 2610 . . . . . . . 8 (.r𝐾) = (.r𝐾)
2321, 22mgpplusg 18316 . . . . . . 7 (.r𝐾) = (+g‘(mulGrp‘𝐾))
242, 23ressplusg 15818 . . . . . 6 ((Unit‘𝐾) ∈ V → (.r𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))))
2520, 24ax-mp 5 . . . . 5 (.r𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))
2625oveqi 6562 . . . 4 (𝑥(.r𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦)
27 fvex 6113 . . . . . 6 (Unit‘𝐿) ∈ V
28 eqid 2610 . . . . . . . 8 (mulGrp‘𝐿) = (mulGrp‘𝐿)
29 eqid 2610 . . . . . . . 8 (.r𝐿) = (.r𝐿)
3028, 29mgpplusg 18316 . . . . . . 7 (.r𝐿) = (+g‘(mulGrp‘𝐿))
3110, 30ressplusg 15818 . . . . . 6 ((Unit‘𝐿) ∈ V → (.r𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
3227, 31ax-mp 5 . . . . 5 (.r𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))
3332oveqi 6562 . . . 4 (𝑥(.r𝐿)𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦)
3419, 26, 333eqtr3g 2667 . . 3 ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦))
354, 12, 34grpinvpropd 17313 . 2 (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))))
36 eqid 2610 . . 3 (invr𝐾) = (invr𝐾)
371, 2, 36invrfval 18496 . 2 (invr𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))
38 eqid 2610 . . 3 (invr𝐿) = (invr𝐿)
399, 10, 38invrfval 18496 . 2 (invr𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))
4035, 37, 393eqtr4g 2669 1 (𝜑 → (invr𝐾) = (invr𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cfv 5804  (class class class)co 6549  Basecbs 15695  s cress 15696  +gcplusg 15768  .rcmulr 15769  invgcminusg 17246  mulGrpcmgp 18312  Unitcui 18462  invrcinvr 18494
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-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-minusg 17249  df-mgp 18313  df-ur 18325  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495
This theorem is referenced by: (None)
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