Theorem crctcshlem4 26712

Description: Lemma for crctcsh 26716. (Contributed by AV, 10-Mar-2021.)

Hypotheses

Ref	Expression
crctcsh.v	|- V = (Vtx ` G )
crctcsh.i	|- I = (iEdg ` G )
crctcsh.d	|- ( ph -> F (Circuits ` G ) P )
crctcsh.n	|- N = ( # ` F )
crctcsh.s	|- ( ph -> S e. ( 0..^ N ) )
crctcsh.h	|- H = ( F cyclShift S )
crctcsh.q	|- Q = ( x e. ( 0 ... N ) |-> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) )

Assertion

Ref	Expression
crctcshlem4	|- ( ( ph /\ S = 0 ) -> ( H = F /\ Q = P ) )

Distinct variable groups:   x, N   x, P   x, S   ph, x

Allowed substitution hints:   Q( x)   F( x)   G( x)   H( x)   I( x)   V( x)

Proof of Theorem crctcshlem4

Step	Hyp	Ref	Expression
1		crctcsh.h	. . 3 |- H = ( F cyclShift S )
2		oveq2 6658	. . . 4 |- ( S = 0 -> ( F cyclShift S ) = ( F cyclShift 0 ) )
3		crctcsh.d	. . . . . 6 |- ( ph -> F (Circuits ` G ) P )
4		crctiswlk 26691	. . . . . 6 |- ( F (Circuits ` G ) P -> F (Walks ` G ) P )
5		crctcsh.i	. . . . . . 7 |- I = (iEdg ` G )
6	5	wlkf 26510	. . . . . 6 |- ( F (Walks ` G ) P -> F e. Word dom I )
7	3, 4, 6	3syl 18	. . . . 5 |- ( ph -> F e. Word dom I )
8		cshw0 13540	. . . . 5 |- ( F e. Word dom I -> ( F cyclShift 0 ) = F )
9	7, 8	syl 17	. . . 4 |- ( ph -> ( F cyclShift 0 ) = F )
10	2, 9	sylan9eqr 2678	. . 3 |- ( ( ph /\ S = 0 ) -> ( F cyclShift S ) = F )
11	1, 10	syl5eq 2668	. 2 |- ( ( ph /\ S = 0 ) -> H = F )
12		crctcsh.q	. . 3 |- Q = ( x e. ( 0 ... N ) |-> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) )
13		oveq2 6658	. . . . . . . . 9 |- ( S = 0 -> ( N - S ) = ( N - 0 ) )
14		crctcsh.v	. . . . . . . . . . . 12 |- V = (Vtx ` G )
15		crctcsh.n	. . . . . . . . . . . 12 |- N = ( # ` F )
16	14, 5, 3, 15	crctcshlem1 26709	. . . . . . . . . . 11 |- ( ph -> N e. NN0 )
17	16	nn0cnd 11353	. . . . . . . . . 10 |- ( ph -> N e. CC )
18	17	subid1d 10381	. . . . . . . . 9 |- ( ph -> ( N - 0 ) = N )
19	13, 18	sylan9eqr 2678	. . . . . . . 8 |- ( ( ph /\ S = 0 ) -> ( N - S ) = N )
20	19	breq2d 4665	. . . . . . 7 |- ( ( ph /\ S = 0 ) -> ( x <_ ( N - S ) <-> x <_ N ) )
21	20	adantr 481	. . . . . 6 |- ( ( ( ph /\ S = 0 ) /\ x e. ( 0 ... N ) ) -> ( x <_ ( N - S ) <-> x <_ N ) )
22		oveq2 6658	. . . . . . . . 9 |- ( S = 0 -> ( x + S ) = ( x + 0 ) )
23	22	adantl 482	. . . . . . . 8 |- ( ( ph /\ S = 0 ) -> ( x + S ) = ( x + 0 ) )
24		elfzelz 12342	. . . . . . . . . 10 |- ( x e. ( 0 ... N ) -> x e. ZZ )
25	24	zcnd 11483	. . . . . . . . 9 |- ( x e. ( 0 ... N ) -> x e. CC )
26	25	addid1d 10236	. . . . . . . 8 |- ( x e. ( 0 ... N ) -> ( x + 0 ) = x )
27	23, 26	sylan9eq 2676	. . . . . . 7 |- ( ( ( ph /\ S = 0 ) /\ x e. ( 0 ... N ) ) -> ( x + S ) = x )
28	27	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ S = 0 ) /\ x e. ( 0 ... N ) ) -> ( P ` ( x + S ) ) = ( P ` x ) )
29	27	oveq1d 6665	. . . . . . 7 |- ( ( ( ph /\ S = 0 ) /\ x e. ( 0 ... N ) ) -> ( ( x + S ) - N ) = ( x - N ) )
30	29	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ S = 0 ) /\ x e. ( 0 ... N ) ) -> ( P ` ( ( x + S ) - N ) ) = ( P ` ( x - N ) ) )
31	21, 28, 30	ifbieq12d 4113	. . . . 5 |- ( ( ( ph /\ S = 0 ) /\ x e. ( 0 ... N ) ) -> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) = if ( x <_ N , ( P ` x ) , ( P ` ( x - N ) ) ) )
32	31	mpteq2dva 4744	. . . 4 |- ( ( ph /\ S = 0 ) -> ( x e. ( 0 ... N ) |-> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) ) = ( x e. ( 0 ... N ) |-> if ( x <_ N , ( P ` x ) , ( P ` ( x - N ) ) ) ) )
33		elfzle2 12345	. . . . . . . . 9 |- ( x e. ( 0 ... N ) -> x <_ N )
34	33	adantl 482	. . . . . . . 8 |- ( ( ph /\ x e. ( 0 ... N ) ) -> x <_ N )
35	34	iftrued 4094	. . . . . . 7 |- ( ( ph /\ x e. ( 0 ... N ) ) -> if ( x <_ N , ( P ` x ) , ( P ` ( x - N ) ) ) = ( P ` x ) )
36	35	mpteq2dva 4744	. . . . . 6 |- ( ph -> ( x e. ( 0 ... N ) |-> if ( x <_ N , ( P ` x ) , ( P ` ( x - N ) ) ) ) = ( x e. ( 0 ... N ) |-> ( P ` x ) ) )
37	14	wlkp 26512	. . . . . . . 8 |- ( F (Walks ` G ) P -> P : ( 0 ... ( # ` F ) ) --> V )
38	3, 4, 37	3syl 18	. . . . . . 7 |- ( ph -> P : ( 0 ... ( # ` F ) ) --> V )
39		ffn 6045	. . . . . . . . . . 11 |- ( P : ( 0 ... ( # ` F ) ) --> V -> P Fn ( 0 ... ( # ` F ) ) )
40	15	eqcomi 2631	. . . . . . . . . . . . 13 |- ( # ` F ) = N
41	40	oveq2i 6661	. . . . . . . . . . . 12 |- ( 0 ... ( # ` F ) ) = ( 0 ... N )
42	41	fneq2i 5986	. . . . . . . . . . 11 |- ( P Fn ( 0 ... ( # ` F ) ) <-> P Fn ( 0 ... N ) )
43	39, 42	sylib 208	. . . . . . . . . 10 |- ( P : ( 0 ... ( # ` F ) ) --> V -> P Fn ( 0 ... N ) )
44	43	adantl 482	. . . . . . . . 9 |- ( ( ph /\ P : ( 0 ... ( # ` F ) ) --> V ) -> P Fn ( 0 ... N ) )
45		dffn5 6241	. . . . . . . . 9 |- ( P Fn ( 0 ... N ) <-> P = ( x e. ( 0 ... N ) |-> ( P ` x ) ) )
46	44, 45	sylib 208	. . . . . . . 8 |- ( ( ph /\ P : ( 0 ... ( # ` F ) ) --> V ) -> P = ( x e. ( 0 ... N ) |-> ( P ` x ) ) )
47	46	eqcomd 2628	. . . . . . 7 |- ( ( ph /\ P : ( 0 ... ( # ` F ) ) --> V ) -> ( x e. ( 0 ... N ) |-> ( P ` x ) ) = P )
48	38, 47	mpdan 702	. . . . . 6 |- ( ph -> ( x e. ( 0 ... N ) |-> ( P ` x ) ) = P )
49	36, 48	eqtrd 2656	. . . . 5 |- ( ph -> ( x e. ( 0 ... N ) |-> if ( x <_ N , ( P ` x ) , ( P ` ( x - N ) ) ) ) = P )
50	49	adantr 481	. . . 4 |- ( ( ph /\ S = 0 ) -> ( x e. ( 0 ... N ) |-> if ( x <_ N , ( P ` x ) , ( P ` ( x - N ) ) ) ) = P )
51	32, 50	eqtrd 2656	. . 3 |- ( ( ph /\ S = 0 ) -> ( x e. ( 0 ... N ) |-> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) ) = P )
52	12, 51	syl5eq 2668	. 2 |- ( ( ph /\ S = 0 ) -> Q = P )
53	11, 52	jca 554	1 |- ( ( ph /\ S = 0 ) -> ( H = F /\ Q = P ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   dom cdm 5114   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   + caddc 9939   <_ cle 10075   - cmin 10266   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   cyclShift ccsh 13534  Vtxcvtx 25874  iEdgciedg 25875  Walkscwlks 26492  Circuitsccrcts 26679

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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  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-int 4476  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-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535  df-wlks 26495  df-trls 26589  df-crcts 26681

This theorem is referenced by:  crctcshwlk  26714  crctcsh  26716