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Theorem unitpropd 18895
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 18893 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
54breq2d 4852 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐾) ↔ 𝑧(∥r𝐾)(1r𝐿)))
64breq2d 4852 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐾) ↔ 𝑧(∥r‘(oppr𝐾))(1r𝐿)))
75, 6anbi12d 618 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿))))
81, 2, 3dvdsrpropd 18894 . . . . . 6 (𝜑 → (∥r𝐾) = (∥r𝐿))
98breqd 4851 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐿) ↔ 𝑧(∥r𝐿)(1r𝐿)))
10 eqid 2805 . . . . . . . . 9 (oppr𝐾) = (oppr𝐾)
11 eqid 2805 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
1210, 11opprbas 18827 . . . . . . . 8 (Base‘𝐾) = (Base‘(oppr𝐾))
131, 12syl6eq 2855 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐾)))
14 eqid 2805 . . . . . . . . 9 (oppr𝐿) = (oppr𝐿)
15 eqid 2805 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1614, 15opprbas 18827 . . . . . . . 8 (Base‘𝐿) = (Base‘(oppr𝐿))
172, 16syl6eq 2855 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐿)))
183ancom2s 632 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
19 eqid 2805 . . . . . . . . 9 (.r𝐾) = (.r𝐾)
20 eqid 2805 . . . . . . . . 9 (.r‘(oppr𝐾)) = (.r‘(oppr𝐾))
2111, 19, 10, 20opprmul 18824 . . . . . . . 8 (𝑦(.r‘(oppr𝐾))𝑥) = (𝑥(.r𝐾)𝑦)
22 eqid 2805 . . . . . . . . 9 (.r𝐿) = (.r𝐿)
23 eqid 2805 . . . . . . . . 9 (.r‘(oppr𝐿)) = (.r‘(oppr𝐿))
2415, 22, 14, 23opprmul 18824 . . . . . . . 8 (𝑦(.r‘(oppr𝐿))𝑥) = (𝑥(.r𝐿)𝑦)
2518, 21, 243eqtr4g 2864 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑦(.r‘(oppr𝐾))𝑥) = (𝑦(.r‘(oppr𝐿))𝑥))
2613, 17, 25dvdsrpropd 18894 . . . . . 6 (𝜑 → (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐿)))
2726breqd 4851 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐿) ↔ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
289, 27anbi12d 618 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
297, 28bitrd 270 . . 3 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
30 eqid 2805 . . . 4 (Unit‘𝐾) = (Unit‘𝐾)
31 eqid 2805 . . . 4 (1r𝐾) = (1r𝐾)
32 eqid 2805 . . . 4 (∥r𝐾) = (∥r𝐾)
33 eqid 2805 . . . 4 (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐾))
3430, 31, 32, 10, 33isunit 18855 . . 3 (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)))
35 eqid 2805 . . . 4 (Unit‘𝐿) = (Unit‘𝐿)
36 eqid 2805 . . . 4 (1r𝐿) = (1r𝐿)
37 eqid 2805 . . . 4 (∥r𝐿) = (∥r𝐿)
38 eqid 2805 . . . 4 (∥r‘(oppr𝐿)) = (∥r‘(oppr𝐿))
3935, 36, 37, 14, 38isunit 18855 . . 3 (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
4029, 34, 393bitr4g 305 . 2 (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿)))
4140eqrdv 2803 1 (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2158   class class class wbr 4840  cfv 6098  (class class class)co 6871  Basecbs 16064  .rcmulr 16150  1rcur 18699  opprcoppr 18820  rcdsr 18836  Unitcui 18837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-rep 4960  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093  ax-un 7176  ax-cnex 10274  ax-resscn 10275  ax-1cn 10276  ax-icn 10277  ax-addcl 10278  ax-addrcl 10279  ax-mulcl 10280  ax-mulrcl 10281  ax-mulcom 10282  ax-addass 10283  ax-mulass 10284  ax-distr 10285  ax-i2m1 10286  ax-1ne0 10287  ax-1rid 10288  ax-rnegex 10289  ax-rrecex 10290  ax-cnre 10291  ax-pre-lttri 10292  ax-pre-lttrn 10293  ax-pre-ltadd 10294  ax-pre-mulgt0 10295
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-reu 3102  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-pss 3782  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4627  df-iun 4710  df-br 4841  df-opab 4903  df-mpt 4920  df-tr 4943  df-id 5216  df-eprel 5221  df-po 5229  df-so 5230  df-fr 5267  df-we 5269  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-pred 5890  df-ord 5936  df-on 5937  df-lim 5938  df-suc 5939  df-iota 6061  df-fun 6100  df-fn 6101  df-f 6102  df-f1 6103  df-fo 6104  df-f1o 6105  df-fv 6106  df-riota 6832  df-ov 6874  df-oprab 6875  df-mpt2 6876  df-om 7293  df-tpos 7584  df-wrecs 7639  df-recs 7701  df-rdg 7739  df-er 7976  df-en 8190  df-dom 8191  df-sdom 8192  df-pnf 10358  df-mnf 10359  df-xr 10360  df-ltxr 10361  df-le 10362  df-sub 10550  df-neg 10551  df-nn 11303  df-2 11360  df-3 11361  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-plusg 16162  df-mulr 16163  df-0g 16303  df-mgp 18688  df-ur 18700  df-oppr 18821  df-dvdsr 18839  df-unit 18840
This theorem is referenced by:  invrpropd  18896  drngprop  18958  drngpropd  18974
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