Theorem submnd0 13028

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 13027	. 2 |- ( ( G e. Mnd /\ .0. e. S /\ S C_ B ) -> .0. = ( 0g ` H ) )
8	1, 2, 3, 7	syl3anc 1249	1 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S ) ) -> .0. = ( 0g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   C_ wss 3154   ` cfv 5255  (class class class)co 5919   Basecbs 12621   ↾_s cress 12622   0gc0g 12870   Mndcmnd 13000

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001

This theorem is referenced by:  subm0  13057