Theorem pmtr3ncomlem1 17893

Description: Lemma 1 for pmtr3ncom 17895. (Contributed by AV, 17-Mar-2018.)

Hypotheses

Ref	Expression
pmtr3ncom.t	|- T = (pmTrsp ` D )
pmtr3ncom.f	|- F = ( T ` { X , Y } )
pmtr3ncom.g	|- G = ( T ` { Y , Z } )

Assertion

Ref	Expression
pmtr3ncomlem1	|- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( G o. F ) ` X ) =/= ( ( F o. G ) ` X ) )

Proof of Theorem pmtr3ncomlem1

Step	Hyp	Ref	Expression
1		necom 2847	. . . . 5 |- ( Y =/= Z <-> Z =/= Y )
2	1	biimpi 206	. . . 4 |- ( Y =/= Z -> Z =/= Y )
3	2	3ad2ant3 1084	. . 3 |- ( ( X =/= Y /\ X =/= Z /\ Y =/= Z ) -> Z =/= Y )
4	3	3ad2ant3 1084	. 2 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Z =/= Y )
5		simp1 1061	. . . . . . . 8 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> D e. V )
6		simp1 1061	. . . . . . . . . 10 |- ( ( X e. D /\ Y e. D /\ Z e. D ) -> X e. D )
7	6	3ad2ant2 1083	. . . . . . . . 9 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> X e. D )
8		simp2 1062	. . . . . . . . . 10 |- ( ( X e. D /\ Y e. D /\ Z e. D ) -> Y e. D )
9	8	3ad2ant2 1083	. . . . . . . . 9 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Y e. D )
10		prssi 4353	. . . . . . . . 9 |- ( ( X e. D /\ Y e. D ) -> { X , Y } C_ D )
11	7, 9, 10	syl2anc 693	. . . . . . . 8 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> { X , Y } C_ D )
12		simp1 1061	. . . . . . . . . . 11 |- ( ( X =/= Y /\ X =/= Z /\ Y =/= Z ) -> X =/= Y )
13	12	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> X =/= Y )
14	7, 9, 13	3jca 1242	. . . . . . . . 9 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( X e. D /\ Y e. D /\ X =/= Y ) )
15		pr2nelem 8827	. . . . . . . . 9 |- ( ( X e. D /\ Y e. D /\ X =/= Y ) -> { X , Y } ~~ 2o )
16	14, 15	syl 17	. . . . . . . 8 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> { X , Y } ~~ 2o )
17	5, 11, 16	3jca 1242	. . . . . . 7 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( D e. V /\ { X , Y } C_ D /\ { X , Y } ~~ 2o ) )
18		pmtr3ncom.t	. . . . . . . 8 |- T = (pmTrsp ` D )
19	18	pmtrf 17875	. . . . . . 7 |- ( ( D e. V /\ { X , Y } C_ D /\ { X , Y } ~~ 2o ) -> ( T ` { X , Y } ) : D --> D )
20	17, 19	syl 17	. . . . . 6 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( T ` { X , Y } ) : D --> D )
21		pmtr3ncom.f	. . . . . . 7 |- F = ( T ` { X , Y } )
22	21	feq1i 6036	. . . . . 6 |- ( F : D --> D <-> ( T ` { X , Y } ) : D --> D )
23	20, 22	sylibr 224	. . . . 5 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> F : D --> D )
24		ffn 6045	. . . . 5 |- ( F : D --> D -> F Fn D )
25	23, 24	syl 17	. . . 4 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> F Fn D )
26		fvco2 6273	. . . 4 |- ( ( F Fn D /\ X e. D ) -> ( ( G o. F ) ` X ) = ( G ` ( F ` X ) ) )
27	25, 7, 26	syl2anc 693	. . 3 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( G o. F ) ` X ) = ( G ` ( F ` X ) ) )
28	21	fveq1i 6192	. . . . 5 |- ( F ` X ) = ( ( T ` { X , Y } ) ` X )
29	18	pmtrprfv 17873	. . . . . 6 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ X =/= Y ) ) -> ( ( T ` { X , Y } ) ` X ) = Y )
30	5, 14, 29	syl2anc 693	. . . . 5 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( T ` { X , Y } ) ` X ) = Y )
31	28, 30	syl5eq 2668	. . . 4 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( F ` X ) = Y )
32	31	fveq2d 6195	. . 3 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( G ` ( F ` X ) ) = ( G ` Y ) )
33		pmtr3ncom.g	. . . . 5 |- G = ( T ` { Y , Z } )
34	33	fveq1i 6192	. . . 4 |- ( G ` Y ) = ( ( T ` { Y , Z } ) ` Y )
35		simp3 1063	. . . . . . 7 |- ( ( X e. D /\ Y e. D /\ Z e. D ) -> Z e. D )
36	35	3ad2ant2 1083	. . . . . 6 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Z e. D )
37		simp3 1063	. . . . . . 7 |- ( ( X =/= Y /\ X =/= Z /\ Y =/= Z ) -> Y =/= Z )
38	37	3ad2ant3 1084	. . . . . 6 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> Y =/= Z )
39	9, 36, 38	3jca 1242	. . . . 5 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( Y e. D /\ Z e. D /\ Y =/= Z ) )
40	18	pmtrprfv 17873	. . . . 5 |- ( ( D e. V /\ ( Y e. D /\ Z e. D /\ Y =/= Z ) ) -> ( ( T ` { Y , Z } ) ` Y ) = Z )
41	5, 39, 40	syl2anc 693	. . . 4 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( T ` { Y , Z } ) ` Y ) = Z )
42	34, 41	syl5eq 2668	. . 3 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( G ` Y ) = Z )
43	27, 32, 42	3eqtrd 2660	. 2 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( G o. F ) ` X ) = Z )
44		prssi 4353	. . . . . . . . 9 |- ( ( Y e. D /\ Z e. D ) -> { Y , Z } C_ D )
45	8, 35, 44	syl2anc 693	. . . . . . . 8 |- ( ( X e. D /\ Y e. D /\ Z e. D ) -> { Y , Z } C_ D )
46	45	3ad2ant2 1083	. . . . . . 7 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> { Y , Z } C_ D )
47		pr2nelem 8827	. . . . . . . 8 |- ( ( Y e. D /\ Z e. D /\ Y =/= Z ) -> { Y , Z } ~~ 2o )
48	39, 47	syl 17	. . . . . . 7 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> { Y , Z } ~~ 2o )
49	5, 46, 48	3jca 1242	. . . . . 6 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( D e. V /\ { Y , Z } C_ D /\ { Y , Z } ~~ 2o ) )
50	18	pmtrf 17875	. . . . . . 7 |- ( ( D e. V /\ { Y , Z } C_ D /\ { Y , Z } ~~ 2o ) -> ( T ` { Y , Z } ) : D --> D )
51	33	feq1i 6036	. . . . . . 7 |- ( G : D --> D <-> ( T ` { Y , Z } ) : D --> D )
52	50, 51	sylibr 224	. . . . . 6 |- ( ( D e. V /\ { Y , Z } C_ D /\ { Y , Z } ~~ 2o ) -> G : D --> D )
53	49, 52	syl 17	. . . . 5 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> G : D --> D )
54		ffn 6045	. . . . 5 |- ( G : D --> D -> G Fn D )
55	53, 54	syl 17	. . . 4 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> G Fn D )
56		fvco2 6273	. . . 4 |- ( ( G Fn D /\ X e. D ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )
57	55, 7, 56	syl2anc 693	. . 3 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( F o. G ) ` X ) = ( F ` ( G ` X ) ) )
58	33	fveq1i 6192	. . . . 5 |- ( G ` X ) = ( ( T ` { Y , Z } ) ` X )
59		id 22	. . . . . 6 |- ( D e. V -> D e. V )
60		3anrot 1043	. . . . . . 7 |- ( ( X e. D /\ Y e. D /\ Z e. D ) <-> ( Y e. D /\ Z e. D /\ X e. D ) )
61	60	biimpi 206	. . . . . 6 |- ( ( X e. D /\ Y e. D /\ Z e. D ) -> ( Y e. D /\ Z e. D /\ X e. D ) )
62		3anrot 1043	. . . . . . 7 |- ( ( Y =/= Z /\ Y =/= X /\ Z =/= X ) <-> ( Y =/= X /\ Z =/= X /\ Y =/= Z ) )
63		necom 2847	. . . . . . . 8 |- ( Y =/= X <-> X =/= Y )
64		necom 2847	. . . . . . . 8 |- ( Z =/= X <-> X =/= Z )
65		biid 251	. . . . . . . 8 |- ( Y =/= Z <-> Y =/= Z )
66	63, 64, 65	3anbi123i 1251	. . . . . . 7 |- ( ( Y =/= X /\ Z =/= X /\ Y =/= Z ) <-> ( X =/= Y /\ X =/= Z /\ Y =/= Z ) )
67	62, 66	sylbbr 226	. . . . . 6 |- ( ( X =/= Y /\ X =/= Z /\ Y =/= Z ) -> ( Y =/= Z /\ Y =/= X /\ Z =/= X ) )
68	18	pmtrprfv3 17874	. . . . . 6 |- ( ( D e. V /\ ( Y e. D /\ Z e. D /\ X e. D ) /\ ( Y =/= Z /\ Y =/= X /\ Z =/= X ) ) -> ( ( T ` { Y , Z } ) ` X ) = X )
69	59, 61, 67, 68	syl3an 1368	. . . . 5 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( T ` { Y , Z } ) ` X ) = X )
70	58, 69	syl5eq 2668	. . . 4 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( G ` X ) = X )
71	70	fveq2d 6195	. . 3 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( F ` ( G ` X ) ) = ( F ` X ) )
72	57, 71, 31	3eqtrd 2660	. 2 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( F o. G ) ` X ) = Y )
73	4, 43, 72	3netr4d 2871	1 |- ( ( D e. V /\ ( X e. D /\ Y e. D /\ Z e. D ) /\ ( X =/= Y /\ X =/= Z /\ Y =/= Z ) ) -> ( ( G o. F ) ` X ) =/= ( ( F o. G ) ` X ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   {cpr 4179   class class class wbr 4653   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888   2oc2o 7554   ~~ cen 7952  pmTrspcpmtr 17861

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

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-ral 2917  df-rex 2918  df-reu 2919  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-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-om 7066  df-1o 7560  df-2o 7561  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pmtr 17862

This theorem is referenced by:  pmtr3ncomlem2  17894