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Theorem unitpropd 19447
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 19445 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
54breq2d 5045 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐾) ↔ 𝑧(∥r𝐾)(1r𝐿)))
64breq2d 5045 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐾) ↔ 𝑧(∥r‘(oppr𝐾))(1r𝐿)))
75, 6anbi12d 633 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿))))
81, 2, 3dvdsrpropd 19446 . . . . . 6 (𝜑 → (∥r𝐾) = (∥r𝐿))
98breqd 5044 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐿) ↔ 𝑧(∥r𝐿)(1r𝐿)))
10 eqid 2801 . . . . . . . . 9 (oppr𝐾) = (oppr𝐾)
11 eqid 2801 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
1210, 11opprbas 19379 . . . . . . . 8 (Base‘𝐾) = (Base‘(oppr𝐾))
131, 12eqtrdi 2852 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐾)))
14 eqid 2801 . . . . . . . . 9 (oppr𝐿) = (oppr𝐿)
15 eqid 2801 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1614, 15opprbas 19379 . . . . . . . 8 (Base‘𝐿) = (Base‘(oppr𝐿))
172, 16eqtrdi 2852 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐿)))
183ancom2s 649 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
19 eqid 2801 . . . . . . . . 9 (.r𝐾) = (.r𝐾)
20 eqid 2801 . . . . . . . . 9 (.r‘(oppr𝐾)) = (.r‘(oppr𝐾))
2111, 19, 10, 20opprmul 19376 . . . . . . . 8 (𝑦(.r‘(oppr𝐾))𝑥) = (𝑥(.r𝐾)𝑦)
22 eqid 2801 . . . . . . . . 9 (.r𝐿) = (.r𝐿)
23 eqid 2801 . . . . . . . . 9 (.r‘(oppr𝐿)) = (.r‘(oppr𝐿))
2415, 22, 14, 23opprmul 19376 . . . . . . . 8 (𝑦(.r‘(oppr𝐿))𝑥) = (𝑥(.r𝐿)𝑦)
2518, 21, 243eqtr4g 2861 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑦(.r‘(oppr𝐾))𝑥) = (𝑦(.r‘(oppr𝐿))𝑥))
2613, 17, 25dvdsrpropd 19446 . . . . . 6 (𝜑 → (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐿)))
2726breqd 5044 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐿) ↔ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
289, 27anbi12d 633 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
297, 28bitrd 282 . . 3 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
30 eqid 2801 . . . 4 (Unit‘𝐾) = (Unit‘𝐾)
31 eqid 2801 . . . 4 (1r𝐾) = (1r𝐾)
32 eqid 2801 . . . 4 (∥r𝐾) = (∥r𝐾)
33 eqid 2801 . . . 4 (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐾))
3430, 31, 32, 10, 33isunit 19407 . . 3 (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)))
35 eqid 2801 . . . 4 (Unit‘𝐿) = (Unit‘𝐿)
36 eqid 2801 . . . 4 (1r𝐿) = (1r𝐿)
37 eqid 2801 . . . 4 (∥r𝐿) = (∥r𝐿)
38 eqid 2801 . . . 4 (∥r‘(oppr𝐿)) = (∥r‘(oppr𝐿))
3935, 36, 37, 14, 38isunit 19407 . . 3 (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
4029, 34, 393bitr4g 317 . 2 (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿)))
4140eqrdv 2799 1 (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112   class class class wbr 5033  cfv 6328  (class class class)co 7139  Basecbs 16479  .rcmulr 16562  1rcur 19248  opprcoppr 19372  rcdsr 19388  Unitcui 19389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-plusg 16574  df-mulr 16575  df-0g 16711  df-mgp 19237  df-ur 19249  df-oppr 19373  df-dvdsr 19391  df-unit 19392
This theorem is referenced by:  invrpropd  19448  drngprop  19510  drngpropd  19526  sradrng  31080
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