Theorem submnd0 13493

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 13492	. 2 |- ( ( G e. Mnd /\ .0. e. S /\ S C_ B ) -> .0. = ( 0g ` H ) )
8	1, 2, 3, 7	syl3anc 1271	1 |- ( ( ( G e. Mnd /\ H e. Mnd ) /\ ( S C_ B /\ .0. e. S ) ) -> .0. = ( 0g ` H ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   C_ wss 3197   ` cfv 5318  (class class class)co 6007   Basecbs 13048   ↾_s cress 13049   0gc0g 13305   Mndcmnd 13465

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-addass 8112  ax-i2m1 8115  ax-0lt1 8116  ax-0id 8118  ax-rnegex 8119  ax-pre-ltirr 8122  ax-pre-ltadd 8126

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-pnf 8194  df-mnf 8195  df-ltxr 8197  df-inn 9122  df-2 9180  df-ndx 13051  df-slot 13052  df-base 13054  df-sets 13055  df-iress 13056  df-plusg 13139  df-0g 13307  df-mgm 13405  df-sgrp 13451  df-mnd 13466

This theorem is referenced by:  subm0  13531