Theorem pj1lid 18114

Description: The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)

Hypotheses

Ref	Expression
pj1eu.a	|- .+ = ( +g ` G )
pj1eu.s	|- .(+) = ( LSSum ` G )
pj1eu.o	|- .0. = ( 0g ` G )
pj1eu.z	|- Z = (Cntz ` G )
pj1eu.2	|- ( ph -> T e. (SubGrp ` G ) )
pj1eu.3	|- ( ph -> U e. (SubGrp ` G ) )
pj1eu.4	|- ( ph -> ( T i^i U ) = { .0. } )
pj1eu.5	|- ( ph -> T C_ ( Z ` U ) )
pj1f.p	|- P = ( proj1 ` G )

Assertion

Ref	Expression
pj1lid	|- ( ( ph /\ X e. T ) -> ( ( T P U ) ` X ) = X )

Proof of Theorem pj1lid

Step	Hyp	Ref	Expression
1		pj1eu.2	. . . . . . 7 |- ( ph -> T e. (SubGrp ` G ) )
2	1	adantr 481	. . . . . 6 |- ( ( ph /\ X e. T ) -> T e. (SubGrp ` G ) )
3		subgrcl 17599	. . . . . 6 |- ( T e. (SubGrp ` G ) -> G e. Grp )
4	2, 3	syl 17	. . . . 5 |- ( ( ph /\ X e. T ) -> G e. Grp )
5		eqid 2622	. . . . . . . 8 |- ( Base ` G ) = ( Base ` G )
6	5	subgss 17595	. . . . . . 7 |- ( T e. (SubGrp ` G ) -> T C_ ( Base ` G ) )
7	1, 6	syl 17	. . . . . 6 |- ( ph -> T C_ ( Base ` G ) )
8	7	sselda 3603	. . . . 5 |- ( ( ph /\ X e. T ) -> X e. ( Base ` G ) )
9		pj1eu.a	. . . . . 6 |- .+ = ( +g ` G )
10		pj1eu.o	. . . . . 6 |- .0. = ( 0g ` G )
11	5, 9, 10	grprid 17453	. . . . 5 |- ( ( G e. Grp /\ X e. ( Base ` G ) ) -> ( X .+ .0. ) = X )
12	4, 8, 11	syl2anc 693	. . . 4 |- ( ( ph /\ X e. T ) -> ( X .+ .0. ) = X )
13	12	eqcomd 2628	. . 3 |- ( ( ph /\ X e. T ) -> X = ( X .+ .0. ) )
14		pj1eu.s	. . . 4 |- .(+) = ( LSSum ` G )
15		pj1eu.z	. . . 4 |- Z = (Cntz ` G )
16		pj1eu.3	. . . . 5 |- ( ph -> U e. (SubGrp ` G ) )
17	16	adantr 481	. . . 4 |- ( ( ph /\ X e. T ) -> U e. (SubGrp ` G ) )
18		pj1eu.4	. . . . 5 |- ( ph -> ( T i^i U ) = { .0. } )
19	18	adantr 481	. . . 4 |- ( ( ph /\ X e. T ) -> ( T i^i U ) = { .0. } )
20		pj1eu.5	. . . . 5 |- ( ph -> T C_ ( Z ` U ) )
21	20	adantr 481	. . . 4 |- ( ( ph /\ X e. T ) -> T C_ ( Z ` U ) )
22		pj1f.p	. . . 4 |- P = ( proj1 ` G )
23	14	lsmub1 18071	. . . . . 6 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> T C_ ( T .(+) U ) )
24	1, 16, 23	syl2anc 693	. . . . 5 |- ( ph -> T C_ ( T .(+) U ) )
25	24	sselda 3603	. . . 4 |- ( ( ph /\ X e. T ) -> X e. ( T .(+) U ) )
26		simpr 477	. . . 4 |- ( ( ph /\ X e. T ) -> X e. T )
27	10	subg0cl 17602	. . . . 5 |- ( U e. (SubGrp ` G ) -> .0. e. U )
28	17, 27	syl 17	. . . 4 |- ( ( ph /\ X e. T ) -> .0. e. U )
29	9, 14, 10, 15, 2, 17, 19, 21, 22, 25, 26, 28	pj1eq 18113	. . 3 |- ( ( ph /\ X e. T ) -> ( X = ( X .+ .0. ) <-> ( ( ( T P U ) ` X ) = X /\ ( ( U P T ) ` X ) = .0. ) ) )
30	13, 29	mpbid 222	. 2 |- ( ( ph /\ X e. T ) -> ( ( ( T P U ) ` X ) = X /\ ( ( U P T ) ` X ) = .0. ) )
31	30	simpld 475	1 |- ( ( ph /\ X e. T ) -> ( ( T P U ) ` X ) = X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Grpcgrp 17422  SubGrpcsubg 17588  Cntzccntz 17748   LSSumclsm 18049   proj1cpj1 18050

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051  df-pj1 18052

This theorem is referenced by:  dpjlid  18460  pjfo  20059