Theorem pj1eu 18109

Description: Uniqueness of a left projection. (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 ) )

Assertion

Ref	Expression
pj1eu	|- ( ( ph /\ X e. ( T .(+) U ) ) -> E! x e. T E. y e. U X = ( x .+ y ) )

Distinct variable groups:   x, y, .+   x, .(+) , y   ph, x, y   x, G, y   x, T, y   x, U, y   x, X, y

Allowed substitution hints:   .0. ( x, y)   Z( x, y)

Proof of Theorem pj1eu

Dummy variables v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pj1eu.2	. . . 4 |- ( ph -> T e. (SubGrp ` G ) )
2		pj1eu.3	. . . 4 |- ( ph -> U e. (SubGrp ` G ) )
3		pj1eu.a	. . . . 5 |- .+ = ( +g ` G )
4		pj1eu.s	. . . . 5 |- .(+) = ( LSSum ` G )
5	3, 4	lsmelval 18064	. . . 4 |- ( ( T e. (SubGrp ` G ) /\ U e. (SubGrp ` G ) ) -> ( X e. ( T .(+) U ) <-> E. x e. T E. y e. U X = ( x .+ y ) ) )
6	1, 2, 5	syl2anc 693	. . 3 |- ( ph -> ( X e. ( T .(+) U ) <-> E. x e. T E. y e. U X = ( x .+ y ) ) )
7	6	biimpa 501	. 2 |- ( ( ph /\ X e. ( T .(+) U ) ) -> E. x e. T E. y e. U X = ( x .+ y ) )
8		reeanv 3107	. . . . 5 |- ( E. y e. U E. v e. U ( X = ( x .+ y ) /\ X = ( u .+ v ) ) <-> ( E. y e. U X = ( x .+ y ) /\ E. v e. U X = ( u .+ v ) ) )
9		eqtr2 2642	. . . . . . 7 |- ( ( X = ( x .+ y ) /\ X = ( u .+ v ) ) -> ( x .+ y ) = ( u .+ v ) )
10		pj1eu.o	. . . . . . . . 9 |- .0. = ( 0g ` G )
11		pj1eu.z	. . . . . . . . 9 |- Z = (Cntz ` G )
12	1	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> T e. (SubGrp ` G ) )
13	2	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> U e. (SubGrp ` G ) )
14		pj1eu.4	. . . . . . . . . 10 |- ( ph -> ( T i^i U ) = { .0. } )
15	14	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> ( T i^i U ) = { .0. } )
16		pj1eu.5	. . . . . . . . . 10 |- ( ph -> T C_ ( Z ` U ) )
17	16	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> T C_ ( Z ` U ) )
18		simplrl 800	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> x e. T )
19		simplrr 801	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> u e. T )
20		simprl 794	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> y e. U )
21		simprr 796	. . . . . . . . 9 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> v e. U )
22	3, 10, 11, 12, 13, 15, 17, 18, 19, 20, 21	subgdisjb 18106	. . . . . . . 8 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> ( ( x .+ y ) = ( u .+ v ) <-> ( x = u /\ y = v ) ) )
23		simpl 473	. . . . . . . 8 |- ( ( x = u /\ y = v ) -> x = u )
24	22, 23	syl6bi 243	. . . . . . 7 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> ( ( x .+ y ) = ( u .+ v ) -> x = u ) )
25	9, 24	syl5 34	. . . . . 6 |- ( ( ( ph /\ ( x e. T /\ u e. T ) ) /\ ( y e. U /\ v e. U ) ) -> ( ( X = ( x .+ y ) /\ X = ( u .+ v ) ) -> x = u ) )
26	25	rexlimdvva 3038	. . . . 5 |- ( ( ph /\ ( x e. T /\ u e. T ) ) -> ( E. y e. U E. v e. U ( X = ( x .+ y ) /\ X = ( u .+ v ) ) -> x = u ) )
27	8, 26	syl5bir 233	. . . 4 |- ( ( ph /\ ( x e. T /\ u e. T ) ) -> ( ( E. y e. U X = ( x .+ y ) /\ E. v e. U X = ( u .+ v ) ) -> x = u ) )
28	27	ralrimivva 2971	. . 3 |- ( ph -> A. x e. T A. u e. T ( ( E. y e. U X = ( x .+ y ) /\ E. v e. U X = ( u .+ v ) ) -> x = u ) )
29	28	adantr 481	. 2 |- ( ( ph /\ X e. ( T .(+) U ) ) -> A. x e. T A. u e. T ( ( E. y e. U X = ( x .+ y ) /\ E. v e. U X = ( u .+ v ) ) -> x = u ) )
30		oveq1 6657	. . . . . 6 |- ( x = u -> ( x .+ y ) = ( u .+ y ) )
31	30	eqeq2d 2632	. . . . 5 |- ( x = u -> ( X = ( x .+ y ) <-> X = ( u .+ y ) ) )
32	31	rexbidv 3052	. . . 4 |- ( x = u -> ( E. y e. U X = ( x .+ y ) <-> E. y e. U X = ( u .+ y ) ) )
33		oveq2 6658	. . . . . 6 |- ( y = v -> ( u .+ y ) = ( u .+ v ) )
34	33	eqeq2d 2632	. . . . 5 |- ( y = v -> ( X = ( u .+ y ) <-> X = ( u .+ v ) ) )
35	34	cbvrexv 3172	. . . 4 |- ( E. y e. U X = ( u .+ y ) <-> E. v e. U X = ( u .+ v ) )
36	32, 35	syl6bb 276	. . 3 |- ( x = u -> ( E. y e. U X = ( x .+ y ) <-> E. v e. U X = ( u .+ v ) ) )
37	36	reu4 3400	. 2 |- ( E! x e. T E. y e. U X = ( x .+ y ) <-> ( E. x e. T E. y e. U X = ( x .+ y ) /\ A. x e. T A. u e. T ( ( E. y e. U X = ( x .+ y ) /\ E. v e. U X = ( u .+ v ) ) -> x = u ) ) )
38	7, 29, 37	sylanbrc 698	1 |- ( ( ph /\ X e. ( T .(+) U ) ) -> E! x e. T E. y e. U X = ( x .+ y ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   i^i cin 3573   C_ wss 3574   {csn 4177   ` cfv 5888  (class class class)co 6650   +g cplusg 15941   0gc0g 16100  SubGrpcsubg 17588  Cntzccntz 17748   LSSumclsm 18049

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-grp 17425  df-minusg 17426  df-sbg 17427  df-subg 17591  df-cntz 17750  df-lsm 18051

This theorem is referenced by:  pj1f  18110  pj1id  18112