Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngcinvALTV Structured version   Visualization version   Unicode version

Theorem rngcinvALTV 40503
 Description: An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
Hypotheses
Ref Expression
rngcsectALTV.c RngCatALTV
rngcsectALTV.b
rngcsectALTV.u
rngcsectALTV.x
rngcsectALTV.y
rngcinvALTV.n Inv
Assertion
Ref Expression
rngcinvALTV RngIsom

Proof of Theorem rngcinvALTV
StepHypRef Expression
1 rngcsectALTV.b . . 3
2 rngcinvALTV.n . . 3 Inv
3 rngcsectALTV.u . . . 4
4 rngcsectALTV.c . . . . 5 RngCatALTV
54rngccatALTV 40500 . . . 4
63, 5syl 17 . . 3
7 rngcsectALTV.x . . 3
8 rngcsectALTV.y . . 3
9 eqid 2471 . . 3 Sect Sect
101, 2, 6, 7, 8, 9isinv 15743 . 2 Sect Sect
11 eqid 2471 . . . . . 6
124, 1, 3, 7, 8, 11, 9rngcsectALTV 40502 . . . . 5 Sect RngHomo RngHomo
13 df-3an 1009 . . . . 5 RngHomo RngHomo RngHomo RngHomo
1412, 13syl6bb 269 . . . 4 Sect RngHomo RngHomo
15 eqid 2471 . . . . . 6
164, 1, 3, 8, 7, 15, 9rngcsectALTV 40502 . . . . 5 Sect RngHomo RngHomo
17 3ancoma 1014 . . . . . 6 RngHomo RngHomo RngHomo RngHomo
18 df-3an 1009 . . . . . 6 RngHomo RngHomo RngHomo RngHomo
1917, 18bitri 257 . . . . 5 RngHomo RngHomo RngHomo RngHomo
2016, 19syl6bb 269 . . . 4 Sect RngHomo RngHomo
2114, 20anbi12d 725 . . 3 Sect Sect RngHomo RngHomo RngHomo RngHomo
22 anandi 844 . . 3 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
2321, 22syl6bb 269 . 2 Sect Sect RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
24 simplrl 778 . . . . . 6 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
2524adantl 473 . . . . 5 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
2611, 15rnghmf 40407 . . . . . . . . . 10 RngHomo
2715, 11rnghmf 40407 . . . . . . . . . 10 RngHomo
2826, 27anim12i 576 . . . . . . . . 9 RngHomo RngHomo
2928ad2antlr 741 . . . . . . . 8 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
30 simpr 468 . . . . . . . . 9 RngHomo RngHomo
3130adantl 473 . . . . . . . 8 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
32 simpr 468 . . . . . . . . 9 RngHomo RngHomo
3332ad2antrl 742 . . . . . . . 8 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
3429, 31, 33jca32 544 . . . . . . 7 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
3534adantl 473 . . . . . 6 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
36 fcof1o 6212 . . . . . . 7
37 eqcom 2478 . . . . . . . 8
3837anbi2i 708 . . . . . . 7
3936, 38sylib 201 . . . . . 6
4035, 39syl 17 . . . . 5 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
41 anass 661 . . . . 5 RngHomo RngHomo
4225, 40, 41sylanbrc 677 . . . 4 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
4311, 15isrngim 40412 . . . . . . 7 RngIsom RngHomo
447, 8, 43syl2anc 673 . . . . . 6 RngIsom RngHomo
4544anbi1d 719 . . . . 5 RngIsom RngHomo
4645adantr 472 . . . 4 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngIsom RngHomo
4742, 46mpbird 240 . . 3 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngIsom
4811, 15rngimrnghm 40414 . . . . . 6 RngIsom RngHomo
4948ad2antrl 742 . . . . 5 RngIsom RngHomo
50 isrngisom 40404 . . . . . . . . . . 11 RngIsom RngHomo RngHomo
517, 8, 50syl2anc 673 . . . . . . . . . 10 RngIsom RngHomo RngHomo
52 eleq1 2537 . . . . . . . . . . . 12 RngHomo RngHomo
5352eqcoms 2479 . . . . . . . . . . 11 RngHomo RngHomo
5453anbi2d 718 . . . . . . . . . 10 RngHomo RngHomo RngHomo RngHomo
5551, 54sylan9bbr 715 . . . . . . . . 9 RngIsom RngHomo RngHomo
56 simpr 468 . . . . . . . . 9 RngHomo RngHomo RngHomo
5755, 56syl6bi 236 . . . . . . . 8 RngIsom RngHomo
5857com12 31 . . . . . . 7 RngIsom RngHomo
5958expdimp 444 . . . . . 6 RngIsom RngHomo
6059impcom 437 . . . . 5 RngIsom RngHomo
61 coeq1 4997 . . . . . . 7
6261ad2antll 743 . . . . . 6 RngIsom
6311, 15rngimf1o 40413 . . . . . . . 8 RngIsom
6463ad2antrl 742 . . . . . . 7 RngIsom
65 f1ococnv1 5856 . . . . . . 7
6664, 65syl 17 . . . . . 6 RngIsom
6762, 66eqtrd 2505 . . . . 5 RngIsom
6849, 60, 67jca31 543 . . . 4 RngIsom RngHomo RngHomo
6951biimpcd 232 . . . . . . 7 RngIsom RngHomo RngHomo
7069adantr 472 . . . . . 6 RngIsom RngHomo RngHomo
7170impcom 437 . . . . 5 RngIsom RngHomo RngHomo
72 eleq1 2537 . . . . . . 7 RngHomo RngHomo
7372ad2antll 743 . . . . . 6 RngIsom RngHomo RngHomo
7473anbi2d 718 . . . . 5 RngIsom RngHomo RngHomo RngHomo RngHomo
7571, 74mpbird 240 . . . 4 RngIsom RngHomo RngHomo
76 coeq2 4998 . . . . . . 7
7776ad2antll 743 . . . . . 6 RngIsom
78 f1ococnv2 5854 . . . . . . 7
7964, 78syl 17 . . . . . 6 RngIsom
8077, 79eqtrd 2505 . . . . 5 RngIsom
8175, 67, 80jca31 543 . . . 4 RngIsom RngHomo RngHomo
8268, 75, 81jca31 543 . . 3 RngIsom RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo
8347, 82impbida 850 . 2 RngHomo RngHomo RngHomo RngHomo RngHomo RngHomo RngIsom
8410, 23, 833bitrd 287 1 RngIsom
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   w3a 1007   wceq 1452   wcel 1904   class class class wbr 4395   cid 4749  ccnv 4838   cres 4841   ccom 4843  wf 5585  wf1o 5588  cfv 5589  (class class class)co 6308  cbs 15199  ccat 15648  Sectcsect 15727  Invcinv 15728   RngHomo crngh 40393   RngIsom crngs 40394  RngCatALTVcrngcALTV 40468 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-hom 15292  df-cco 15293  df-0g 15418  df-cat 15652  df-cid 15653  df-sect 15730  df-inv 15731  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-mhm 16660  df-grp 16751  df-ghm 16959  df-abl 17511  df-mgp 17802  df-mgmhm 40287  df-rng0 40383  df-rnghomo 40395  df-rngisom 40396  df-rngcALTV 40470 This theorem is referenced by:  rngcisoALTV  40504
 Copyright terms: Public domain W3C validator