Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sectmon Structured version   Unicode version

Theorem sectmon 14008
 Description: If is a section of , then is a monomorphism. A monomorphism that arises from a section is also known as a split monomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
sectmon.b
sectmon.m Mono
sectmon.s Sect
sectmon.c
sectmon.x
sectmon.y
sectmon.1
Assertion
Ref Expression
sectmon

Proof of Theorem sectmon
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sectmon.1 . . . 4
2 sectmon.b . . . . 5
3 eqid 2438 . . . . 5
4 eqid 2438 . . . . 5 comp comp
5 eqid 2438 . . . . 5
6 sectmon.s . . . . 5 Sect
7 sectmon.c . . . . 5
8 sectmon.x . . . . 5
9 sectmon.y . . . . 5
102, 3, 4, 5, 6, 7, 8, 9issect 13984 . . . 4 comp
111, 10mpbid 203 . . 3 comp
1211simp1d 970 . 2
13 oveq2 6092 . . . . 5 comp comp comp comp comp comp
1411simp3d 972 . . . . . . . . 9 comp
1514ad2antrr 708 . . . . . . . 8 comp
1615oveq1d 6099 . . . . . . 7 comp comp comp
177ad2antrr 708 . . . . . . . 8
18 simplr 733 . . . . . . . 8
198ad2antrr 708 . . . . . . . 8
209ad2antrr 708 . . . . . . . 8
21 simprl 734 . . . . . . . 8
2212ad2antrr 708 . . . . . . . 8
2311simp2d 971 . . . . . . . . 9
2423ad2antrr 708 . . . . . . . 8
252, 3, 4, 17, 18, 19, 20, 21, 22, 19, 24catass 13916 . . . . . . 7 comp comp comp comp
262, 3, 5, 17, 18, 4, 19, 21catlid 13913 . . . . . . 7 comp
2716, 25, 263eqtr3d 2478 . . . . . 6 comp comp
2815oveq1d 6099 . . . . . . 7 comp comp comp
29 simprr 735 . . . . . . . 8
302, 3, 4, 17, 18, 19, 20, 29, 22, 19, 24catass 13916 . . . . . . 7 comp comp comp comp
312, 3, 5, 17, 18, 4, 19, 29catlid 13913 . . . . . . 7 comp
3228, 30, 313eqtr3d 2478 . . . . . 6 comp comp
3327, 32eqeq12d 2452 . . . . 5 comp comp comp comp
3413, 33syl5ib 212 . . . 4 comp comp
3534ralrimivva 2800 . . 3 comp comp
3635ralrimiva 2791 . 2 comp comp
37 sectmon.m . . 3 Mono
382, 3, 4, 37, 7, 8, 9ismon2 13965 . 2 comp comp
3912, 36, 38mpbir2and 890 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  cop 3819   class class class wbr 4215  cfv 5457  (class class class)co 6084  cbs 13474   chom 13545  compcco 13546  ccat 13894  ccid 13895  Monocmon 13959  Sectcsect 13975 This theorem is referenced by:  sectepi  14010 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-cat 13898  df-cid 13899  df-mon 13961  df-sect 13978
 Copyright terms: Public domain W3C validator