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Theorem unitmulcl 15769
 Description: The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
unitmulcl.1 Unit
unitmulcl.2
Assertion
Ref Expression
unitmulcl

Proof of Theorem unitmulcl
StepHypRef Expression
1 simp1 957 . . 3
2 simp3 959 . . . . . 6
3 eqid 2436 . . . . . . 7
4 unitmulcl.1 . . . . . . 7 Unit
53, 4unitcl 15764 . . . . . 6
62, 5syl 16 . . . . 5
7 simp2 958 . . . . . . 7
8 eqid 2436 . . . . . . . 8
9 eqid 2436 . . . . . . . 8 r r
10 eqid 2436 . . . . . . . 8 oppr oppr
11 eqid 2436 . . . . . . . 8 roppr roppr
124, 8, 9, 10, 11isunit 15762 . . . . . . 7 r roppr
137, 12sylib 189 . . . . . 6 r roppr
1413simpld 446 . . . . 5 r
15 unitmulcl.2 . . . . . 6
163, 9, 15dvdsrmul1 15758 . . . . 5 r r
171, 6, 14, 16syl3anc 1184 . . . 4 r
183, 15, 8rnglidm 15687 . . . . 5
191, 6, 18syl2anc 643 . . . 4
2017, 19breqtrd 4236 . . 3 r
214, 8, 9, 10, 11isunit 15762 . . . . 5 r roppr
222, 21sylib 189 . . . 4 r roppr
2322simpld 446 . . 3 r
243, 9dvdsrtr 15757 . . 3 r r r
251, 20, 23, 24syl3anc 1184 . 2 r
2610opprrng 15736 . . . 4 oppr
271, 26syl 16 . . 3 oppr
28 eqid 2436 . . . . 5 oppr oppr
293, 15, 10, 28opprmul 15731 . . . 4 oppr
303, 4unitcl 15764 . . . . . . 7
317, 30syl 16 . . . . . 6
3222simprd 450 . . . . . 6 roppr
3310, 3opprbas 15734 . . . . . . 7 oppr
3433, 11, 28dvdsrmul1 15758 . . . . . 6 oppr roppr opprropproppr
3527, 31, 32, 34syl3anc 1184 . . . . 5 opprropproppr
363, 15, 10, 28opprmul 15731 . . . . . 6 oppr
373, 15, 8rngridm 15688 . . . . . . 7
381, 31, 37syl2anc 643 . . . . . 6
3936, 38syl5eq 2480 . . . . 5 oppr
4035, 39breqtrd 4236 . . . 4 opprroppr
4129, 40syl5eqbrr 4246 . . 3 roppr
4213simprd 450 . . 3 roppr
4333, 11dvdsrtr 15757 . . 3 oppr roppr roppr roppr
4427, 41, 42, 43syl3anc 1184 . 2 roppr
454, 8, 9, 10, 11isunit 15762 . 2 r roppr
4625, 44, 45sylanbrc 646 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725   class class class wbr 4212  cfv 5454  (class class class)co 6081  cbs 13469  cmulr 13530  crg 15660  cur 15662  opprcoppr 15727  rcdsr 15743  Unitcui 15744 This theorem is referenced by:  unitmulclb  15770  unitgrp  15772  unitdvcl  15792  irredrmul  15812  subrgugrp  15887  dchrelbasd  21023  dchrptlem2  21049  rdivmuldivd  24227  dvrcan5  24229  qqhghm  24372  qqhrhm  24373 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-plusg 13542  df-mulr 13543  df-0g 13727  df-mnd 14690  df-grp 14812  df-mgp 15649  df-rng 15663  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747
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