subg0cl - Metamath Proof Explorer

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Theorem subg0cl 19089

Description: The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)

**Hypothesis**
Ref	Expression
subg0cl.i	⊢ 0 = (0_g‘𝐺)

**Assertion**
Ref	Expression
subg0cl	⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)

**Proof of Theorem subg0cl**
Step	Hyp	Ref	Expression
1		eqid 2728	. . . 4 ⊢ (𝐺 ↾_s 𝑆) = (𝐺 ↾_s 𝑆)
2	1	subggrp 19084	. . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾_s 𝑆) ∈ Grp)
3		eqid 2728	. . . 4 ⊢ (Base‘(𝐺 ↾_s 𝑆)) = (Base‘(𝐺 ↾_s 𝑆))
4		eqid 2728	. . . 4 ⊢ (0_g‘(𝐺 ↾_s 𝑆)) = (0_g‘(𝐺 ↾_s 𝑆))
5	3, 4	grpidcl 18922	. . 3 ⊢ ((𝐺 ↾_s 𝑆) ∈ Grp → (0_g‘(𝐺 ↾_s 𝑆)) ∈ (Base‘(𝐺 ↾_s 𝑆)))
6	2, 5	syl 17	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0_g‘(𝐺 ↾_s 𝑆)) ∈ (Base‘(𝐺 ↾_s 𝑆)))
7		subg0cl.i	. . 3 ⊢ 0 = (0_g‘𝐺)
8	1, 7	subg0 19087	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0_g‘(𝐺 ↾_s 𝑆)))
9	1	subgbas 19085	. 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺 ↾_s 𝑆)))
10	6, 8, 9	3eltr4d 2844	1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 ↾_s cress 17209 0_gc0g 17421 Grpcgrp 18890 SubGrpcsubg 19075

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 df-subg 19078

This theorem is referenced by: subgmulgcl 19094 issubg3 19099 issubg4 19100 subgint 19105 trivsubgd 19108 eqger 19133 ghmpreima 19192 subgga 19251 gasubg 19253 sylow1lem5 19557 sylow2blem2 19576 sylow2blem3 19577 fislw 19580 sylow3lem3 19584 sylow3lem4 19585 lsm01 19626 lsm02 19627 lsmdisj 19636 lsmdisj2 19637 pj1lid 19656 pj1rid 19657 dmdprdd 19956 dprdfid 19974 dprdfeq0 19979 dprdsubg 19981 dprdres 19985 dprdz 19987 dprdsn 19993 dmdprdsplitlem 19994 dprddisj2 19996 dprd2da 19999 dmdprdsplit2lem 20002 ablfacrp 20023 ablfacrp2 20024 ablfac1c 20028 ablfac1eu 20030 pgpfac1lem3a 20033 pgpfac1lem3 20034 pgpfac1lem5 20036 pgpfaclem2 20039 pgpfaclem3 20040 prmgrpsimpgd 20071 primefld0cl 20694 abvres 20719 islss4 20846 dflidl2rng 21114 rnglidlrng 21142 rng2idl0 21161 rng2idlsubg0 21164 2idlcpblrng 21165 rng2idl1cntr 21195 subrgpsr 21921 mpllsslem 21942 0elcpmat 22637 opnsubg 24025 clssubg 24026 tgpconncompss 24031 plypf1 26159 dvply2g 26232 dvply2gOLD 26233 efsubm 26498 dchrptlem3 27212 gsumsubg 32773 nsgqus0 33133 nsgqusf1olem1 33136 ressply10g 33252 ressply1invg 33254 drgext0gsca 33291 fedgmullem2 33328 algextdeglem4 33388 algextdeglem5 33389 fsumcnsrcl 42590 cnsrplycl 42591 rngunsnply 42597

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