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Theorem unitpropd 18818
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 18816 . . . . . 6 (𝜑 → (1r𝐾) = (1r𝐿))
54breq2d 4772 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐾) ↔ 𝑧(∥r𝐾)(1r𝐿)))
64breq2d 4772 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐾) ↔ 𝑧(∥r‘(oppr𝐾))(1r𝐿)))
75, 6anbi12d 749 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿))))
81, 2, 3dvdsrpropd 18817 . . . . . 6 (𝜑 → (∥r𝐾) = (∥r𝐿))
98breqd 4771 . . . . 5 (𝜑 → (𝑧(∥r𝐾)(1r𝐿) ↔ 𝑧(∥r𝐿)(1r𝐿)))
10 eqid 2724 . . . . . . . . 9 (oppr𝐾) = (oppr𝐾)
11 eqid 2724 . . . . . . . . 9 (Base‘𝐾) = (Base‘𝐾)
1210, 11opprbas 18750 . . . . . . . 8 (Base‘𝐾) = (Base‘(oppr𝐾))
131, 12syl6eq 2774 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐾)))
14 eqid 2724 . . . . . . . . 9 (oppr𝐿) = (oppr𝐿)
15 eqid 2724 . . . . . . . . 9 (Base‘𝐿) = (Base‘𝐿)
1614, 15opprbas 18750 . . . . . . . 8 (Base‘𝐿) = (Base‘(oppr𝐿))
172, 16syl6eq 2774 . . . . . . 7 (𝜑𝐵 = (Base‘(oppr𝐿)))
183ancom2s 879 . . . . . . . 8 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))
19 eqid 2724 . . . . . . . . 9 (.r𝐾) = (.r𝐾)
20 eqid 2724 . . . . . . . . 9 (.r‘(oppr𝐾)) = (.r‘(oppr𝐾))
2111, 19, 10, 20opprmul 18747 . . . . . . . 8 (𝑦(.r‘(oppr𝐾))𝑥) = (𝑥(.r𝐾)𝑦)
22 eqid 2724 . . . . . . . . 9 (.r𝐿) = (.r𝐿)
23 eqid 2724 . . . . . . . . 9 (.r‘(oppr𝐿)) = (.r‘(oppr𝐿))
2415, 22, 14, 23opprmul 18747 . . . . . . . 8 (𝑦(.r‘(oppr𝐿))𝑥) = (𝑥(.r𝐿)𝑦)
2518, 21, 243eqtr4g 2783 . . . . . . 7 ((𝜑 ∧ (𝑦𝐵𝑥𝐵)) → (𝑦(.r‘(oppr𝐾))𝑥) = (𝑦(.r‘(oppr𝐿))𝑥))
2613, 17, 25dvdsrpropd 18817 . . . . . 6 (𝜑 → (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐿)))
2726breqd 4771 . . . . 5 (𝜑 → (𝑧(∥r‘(oppr𝐾))(1r𝐿) ↔ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
289, 27anbi12d 749 . . . 4 (𝜑 → ((𝑧(∥r𝐾)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐿)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
297, 28bitrd 268 . . 3 (𝜑 → ((𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿))))
30 eqid 2724 . . . 4 (Unit‘𝐾) = (Unit‘𝐾)
31 eqid 2724 . . . 4 (1r𝐾) = (1r𝐾)
32 eqid 2724 . . . 4 (∥r𝐾) = (∥r𝐾)
33 eqid 2724 . . . 4 (∥r‘(oppr𝐾)) = (∥r‘(oppr𝐾))
3430, 31, 32, 10, 33isunit 18778 . . 3 (𝑧 ∈ (Unit‘𝐾) ↔ (𝑧(∥r𝐾)(1r𝐾) ∧ 𝑧(∥r‘(oppr𝐾))(1r𝐾)))
35 eqid 2724 . . . 4 (Unit‘𝐿) = (Unit‘𝐿)
36 eqid 2724 . . . 4 (1r𝐿) = (1r𝐿)
37 eqid 2724 . . . 4 (∥r𝐿) = (∥r𝐿)
38 eqid 2724 . . . 4 (∥r‘(oppr𝐿)) = (∥r‘(oppr𝐿))
3935, 36, 37, 14, 38isunit 18778 . . 3 (𝑧 ∈ (Unit‘𝐿) ↔ (𝑧(∥r𝐿)(1r𝐿) ∧ 𝑧(∥r‘(oppr𝐿))(1r𝐿)))
4029, 34, 393bitr4g 303 . 2 (𝜑 → (𝑧 ∈ (Unit‘𝐾) ↔ 𝑧 ∈ (Unit‘𝐿)))
4140eqrdv 2722 1 (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103   class class class wbr 4760  cfv 6001  (class class class)co 6765  Basecbs 15980  .rcmulr 16065  1rcur 18622  opprcoppr 18743  rcdsr 18759  Unitcui 18760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-tpos 7472  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-er 7862  df-en 8073  df-dom 8074  df-sdom 8075  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-2 11192  df-3 11193  df-ndx 15983  df-slot 15984  df-base 15986  df-sets 15987  df-plusg 16077  df-mulr 16078  df-0g 16225  df-mgp 18611  df-ur 18623  df-oppr 18744  df-dvdsr 18762  df-unit 18763
This theorem is referenced by:  invrpropd  18819  drngprop  18881  drngpropd  18897
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