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Mirrors > Home > MPE Home > Th. List > subm0cl | Structured version Visualization version GIF version |
Description: Submonoids contain zero. (Contributed by Mario Carneiro, 7-Mar-2015.) |
Ref | Expression |
---|---|
subm0cl.z | ⊢ 0 = (0g‘𝑀) |
Ref | Expression |
---|---|
subm0cl | ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 0 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | submrcl 17961 | . . . 4 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 𝑀 ∈ Mnd) | |
2 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑀) = (Base‘𝑀) | |
3 | subm0cl.z | . . . . 5 ⊢ 0 = (0g‘𝑀) | |
4 | eqid 2821 | . . . . 5 ⊢ (𝑀 ↾s 𝑆) = (𝑀 ↾s 𝑆) | |
5 | 2, 3, 4 | issubm2 17963 | . . . 4 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ 0 ∈ 𝑆 ∧ (𝑀 ↾s 𝑆) ∈ Mnd))) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ 0 ∈ 𝑆 ∧ (𝑀 ↾s 𝑆) ∈ Mnd))) |
7 | 6 | ibi 269 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → (𝑆 ⊆ (Base‘𝑀) ∧ 0 ∈ 𝑆 ∧ (𝑀 ↾s 𝑆) ∈ Mnd)) |
8 | 7 | simp2d 1139 | 1 ⊢ (𝑆 ∈ (SubMnd‘𝑀) → 0 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 ↾s cress 16478 0gc0g 16707 Mndcmnd 17905 SubMndcsubmnd 17949 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 |
This theorem is referenced by: subm0 17974 subsubm 17975 resmhm 17979 mhmima 17983 gsumsubm 17993 gsumwsubmcl 17995 submmulgcl 18264 issubg3 18291 symggen 18592 lsmub1x 18765 lsmub2x 18766 lsmsubm 18772 gsumzsplit 19041 gsumpt 19076 chfacfisfcpmat 21457 |
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