Theorem submnd0 13276

Description: The zero of a submonoid is the same as the zero in the parent monoid. (Note that we must add the condition that the zero of the parent monoid is actually contained in the submonoid, because it is possible to have "subsets that are monoids" which are not submonoids because they have a different identity element. (Contributed by Mario Carneiro, 10-Jan-2015.)

Hypotheses

Ref	Expression
submnd0.b	|- B = ( Base ` G )
submnd0.z	|- .0. = ( 0g ` G )
submnd0.h	|- H = ( G ↾s S )

Assertion

Ref	Expression
submnd0	|- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S ) ) -> .0. = ( 0g ` H ) )

Proof of Theorem submnd0

Step	Hyp	Ref	Expression
1		simpll 527	. 2 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S ) ) -> G e. Mnd )
2		simprr 531	. 2 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S ) ) -> .0. e. S )
3		simprl 529	. 2 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S ) ) -> S C_ B )
4		submnd0.h	. . 3 |- H = ( G ↾s S )
5		submnd0.b	. . 3 |- B = ( Base ` G )
6		submnd0.z	. . 3 |- .0. = ( 0g ` G )
7	4, 5, 6	ress0g 13275	. 2 |- ( ( G e. Mnd /\ .0. e. S /\ S C_ B ) -> .0. = ( 0g ` H ) )
8	1, 2, 3, 7	syl3anc 1250	1 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S ) ) -> .0. = ( 0g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   C_ wss 3166   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833   0gc0g 13088   Mndcmnd 13248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249

This theorem is referenced by:  subm0  13314