Theorem crctcsh 26716

Description: Cyclically shifting the indices of a circuit <. F , P >. results in a circuit <. H , Q >.. (Contributed by AV, 10-Mar-2021.) (Proof shortened by AV, 31-Oct-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
crctcsh	|- ( ph -> H (Circuits ` G ) Q )

Distinct variable groups:   x, N   x, P   x, S   ph, x   x, F   x, I   x, V   x, H

Allowed substitution hints:   Q( x)   G( x)

Proof of Theorem crctcsh

Step	Hyp	Ref	Expression
1		crctcsh.v	. . . 4 |- V = (Vtx ` G )
2		crctcsh.i	. . . 4 |- I = (iEdg ` G )
3		crctcsh.d	. . . 4 |- ( ph -> F (Circuits ` G ) P )
4		crctcsh.n	. . . 4 |- N = ( # ` F )
5		crctcsh.s	. . . 4 |- ( ph -> S e. ( 0..^ N ) )
6		crctcsh.h	. . . 4 |- H = ( F cyclShift S )
7		crctcsh.q	. . . 4 |- Q = ( x e. ( 0 ... N ) |-> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) )
8	1, 2, 3, 4, 5, 6, 7	crctcshlem4 26712	. . 3 |- ( ( ph /\ S = 0 ) -> ( H = F /\ Q = P ) )
9		breq12 4658	. . . . 5 |- ( ( H = F /\ Q = P ) -> ( H (Circuits ` G ) Q <-> F (Circuits ` G ) P ) )
10	3, 9	syl5ibrcom 237	. . . 4 |- ( ph -> ( ( H = F /\ Q = P ) -> H (Circuits ` G ) Q ) )
11	10	adantr 481	. . 3 |- ( ( ph /\ S = 0 ) -> ( ( H = F /\ Q = P ) -> H (Circuits ` G ) Q ) )
12	8, 11	mpd 15	. 2 |- ( ( ph /\ S = 0 ) -> H (Circuits ` G ) Q )
13	1, 2, 3, 4, 5, 6, 7	crctcshtrl 26715	. . . 4 |- ( ph -> H (Trails ` G ) Q )
14	13	adantr 481	. . 3 |- ( ( ph /\ S =/= 0 ) -> H (Trails ` G ) Q )
15	7	a1i 11	. . . . 5 |- ( ( ph /\ S =/= 0 ) -> Q = ( x e. ( 0 ... N ) |-> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) ) )
16		breq1 4656	. . . . . . 7 |- ( x = 0 -> ( x <_ ( N - S ) <-> 0 <_ ( N - S ) ) )
17		oveq1 6657	. . . . . . . 8 |- ( x = 0 -> ( x + S ) = ( 0 + S ) )
18	17	fveq2d 6195	. . . . . . 7 |- ( x = 0 -> ( P ` ( x + S ) ) = ( P ` ( 0 + S ) ) )
19	17	oveq1d 6665	. . . . . . . 8 |- ( x = 0 -> ( ( x + S ) - N ) = ( ( 0 + S ) - N ) )
20	19	fveq2d 6195	. . . . . . 7 |- ( x = 0 -> ( P ` ( ( x + S ) - N ) ) = ( P ` ( ( 0 + S ) - N ) ) )
21	16, 18, 20	ifbieq12d 4113	. . . . . 6 |- ( x = 0 -> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) = if ( 0 <_ ( N - S ) , ( P ` ( 0 + S ) ) , ( P ` ( ( 0 + S ) - N ) ) ) )
22		elfzo0le 12511	. . . . . . . . . 10 |- ( S e. ( 0..^ N ) -> S <_ N )
23	5, 22	syl 17	. . . . . . . . 9 |- ( ph -> S <_ N )
24	1, 2, 3, 4	crctcshlem1 26709	. . . . . . . . . . 11 |- ( ph -> N e. NN0 )
25	24	nn0red 11352	. . . . . . . . . 10 |- ( ph -> N e. RR )
26		elfzoelz 12470	. . . . . . . . . . . 12 |- ( S e. ( 0..^ N ) -> S e. ZZ )
27	5, 26	syl 17	. . . . . . . . . . 11 |- ( ph -> S e. ZZ )
28	27	zred 11482	. . . . . . . . . 10 |- ( ph -> S e. RR )
29	25, 28	subge0d 10617	. . . . . . . . 9 |- ( ph -> ( 0 <_ ( N - S ) <-> S <_ N ) )
30	23, 29	mpbird 247	. . . . . . . 8 |- ( ph -> 0 <_ ( N - S ) )
31	30	adantr 481	. . . . . . 7 |- ( ( ph /\ S =/= 0 ) -> 0 <_ ( N - S ) )
32	31	iftrued 4094	. . . . . 6 |- ( ( ph /\ S =/= 0 ) -> if ( 0 <_ ( N - S ) , ( P ` ( 0 + S ) ) , ( P ` ( ( 0 + S ) - N ) ) ) = ( P ` ( 0 + S ) ) )
33	21, 32	sylan9eqr 2678	. . . . 5 |- ( ( ( ph /\ S =/= 0 ) /\ x = 0 ) -> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) = ( P ` ( 0 + S ) ) )
34	3	adantr 481	. . . . . . 7 |- ( ( ph /\ S =/= 0 ) -> F (Circuits ` G ) P )
35	1, 2, 34, 4	crctcshlem1 26709	. . . . . 6 |- ( ( ph /\ S =/= 0 ) -> N e. NN0 )
36		0elfz 12436	. . . . . 6 |- ( N e. NN0 -> 0 e. ( 0 ... N ) )
37	35, 36	syl 17	. . . . 5 |- ( ( ph /\ S =/= 0 ) -> 0 e. ( 0 ... N ) )
38		fvexd 6203	. . . . 5 |- ( ( ph /\ S =/= 0 ) -> ( P ` ( 0 + S ) ) e. _V )
39	15, 33, 37, 38	fvmptd 6288	. . . 4 |- ( ( ph /\ S =/= 0 ) -> ( Q ` 0 ) = ( P ` ( 0 + S ) ) )
40		breq1 4656	. . . . . . . 8 |- ( x = ( # ` H ) -> ( x <_ ( N - S ) <-> ( # ` H ) <_ ( N - S ) ) )
41		oveq1 6657	. . . . . . . . 9 |- ( x = ( # ` H ) -> ( x + S ) = ( ( # ` H ) + S ) )
42	41	fveq2d 6195	. . . . . . . 8 |- ( x = ( # ` H ) -> ( P ` ( x + S ) ) = ( P ` ( ( # ` H ) + S ) ) )
43	41	oveq1d 6665	. . . . . . . . 9 |- ( x = ( # ` H ) -> ( ( x + S ) - N ) = ( ( ( # ` H ) + S ) - N ) )
44	43	fveq2d 6195	. . . . . . . 8 |- ( x = ( # ` H ) -> ( P ` ( ( x + S ) - N ) ) = ( P ` ( ( ( # ` H ) + S ) - N ) ) )
45	40, 42, 44	ifbieq12d 4113	. . . . . . 7 |- ( x = ( # ` H ) -> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) = if ( ( # ` H ) <_ ( N - S ) , ( P ` ( ( # ` H ) + S ) ) , ( P ` ( ( ( # ` H ) + S ) - N ) ) ) )
46		elfzoel2 12469	. . . . . . . . . . . 12 |- ( S e. ( 0..^ N ) -> N e. ZZ )
47		elfzonn0 12512	. . . . . . . . . . . 12 |- ( S e. ( 0..^ N ) -> S e. NN0 )
48		simpr 477	. . . . . . . . . . . . . . . . 17 |- ( ( N e. ZZ /\ S e. NN0 ) -> S e. NN0 )
49	48	anim1i 592	. . . . . . . . . . . . . . . 16 |- ( ( ( N e. ZZ /\ S e. NN0 ) /\ S =/= 0 ) -> ( S e. NN0 /\ S =/= 0 ) )
50		elnnne0 11306	. . . . . . . . . . . . . . . 16 |- ( S e. NN <-> ( S e. NN0 /\ S =/= 0 ) )
51	49, 50	sylibr 224	. . . . . . . . . . . . . . 15 |- ( ( ( N e. ZZ /\ S e. NN0 ) /\ S =/= 0 ) -> S e. NN )
52	51	nngt0d 11064	. . . . . . . . . . . . . 14 |- ( ( ( N e. ZZ /\ S e. NN0 ) /\ S =/= 0 ) -> 0 < S )
53		zre 11381	. . . . . . . . . . . . . . . . 17 |- ( N e. ZZ -> N e. RR )
54		nn0re 11301	. . . . . . . . . . . . . . . . 17 |- ( S e. NN0 -> S e. RR )
55	53, 54	anim12ci 591	. . . . . . . . . . . . . . . 16 |- ( ( N e. ZZ /\ S e. NN0 ) -> ( S e. RR /\ N e. RR ) )
56	55	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( N e. ZZ /\ S e. NN0 ) /\ S =/= 0 ) -> ( S e. RR /\ N e. RR ) )
57		ltsubpos 10520	. . . . . . . . . . . . . . . 16 |- ( ( S e. RR /\ N e. RR ) -> ( 0 < S <-> ( N - S ) < N ) )
58	57	bicomd 213	. . . . . . . . . . . . . . 15 |- ( ( S e. RR /\ N e. RR ) -> ( ( N - S ) < N <-> 0 < S ) )
59	56, 58	syl 17	. . . . . . . . . . . . . 14 |- ( ( ( N e. ZZ /\ S e. NN0 ) /\ S =/= 0 ) -> ( ( N - S ) < N <-> 0 < S ) )
60	52, 59	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( N e. ZZ /\ S e. NN0 ) /\ S =/= 0 ) -> ( N - S ) < N )
61	60	ex 450	. . . . . . . . . . . 12 |- ( ( N e. ZZ /\ S e. NN0 ) -> ( S =/= 0 -> ( N - S ) < N ) )
62	46, 47, 61	syl2anc 693	. . . . . . . . . . 11 |- ( S e. ( 0..^ N ) -> ( S =/= 0 -> ( N - S ) < N ) )
63	5, 62	syl 17	. . . . . . . . . 10 |- ( ph -> ( S =/= 0 -> ( N - S ) < N ) )
64	63	imp 445	. . . . . . . . 9 |- ( ( ph /\ S =/= 0 ) -> ( N - S ) < N )
65	5	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ S =/= 0 ) -> S e. ( 0..^ N ) )
66	1, 2, 34, 4, 65, 6	crctcshlem2 26710	. . . . . . . . . . . 12 |- ( ( ph /\ S =/= 0 ) -> ( # ` H ) = N )
67	66	breq1d 4663	. . . . . . . . . . 11 |- ( ( ph /\ S =/= 0 ) -> ( ( # ` H ) <_ ( N - S ) <-> N <_ ( N - S ) ) )
68	67	notbid 308	. . . . . . . . . 10 |- ( ( ph /\ S =/= 0 ) -> ( -. ( # ` H ) <_ ( N - S ) <-> -. N <_ ( N - S ) ) )
69	25, 28	resubcld 10458	. . . . . . . . . . . . 13 |- ( ph -> ( N - S ) e. RR )
70	69, 25	jca 554	. . . . . . . . . . . 12 |- ( ph -> ( ( N - S ) e. RR /\ N e. RR ) )
71	70	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ S =/= 0 ) -> ( ( N - S ) e. RR /\ N e. RR ) )
72		ltnle 10117	. . . . . . . . . . 11 |- ( ( ( N - S ) e. RR /\ N e. RR ) -> ( ( N - S ) < N <-> -. N <_ ( N - S ) ) )
73	71, 72	syl 17	. . . . . . . . . 10 |- ( ( ph /\ S =/= 0 ) -> ( ( N - S ) < N <-> -. N <_ ( N - S ) ) )
74	68, 73	bitr4d 271	. . . . . . . . 9 |- ( ( ph /\ S =/= 0 ) -> ( -. ( # ` H ) <_ ( N - S ) <-> ( N - S ) < N ) )
75	64, 74	mpbird 247	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> -. ( # ` H ) <_ ( N - S ) )
76	75	iffalsed 4097	. . . . . . 7 |- ( ( ph /\ S =/= 0 ) -> if ( ( # ` H ) <_ ( N - S ) , ( P ` ( ( # ` H ) + S ) ) , ( P ` ( ( ( # ` H ) + S ) - N ) ) ) = ( P ` ( ( ( # ` H ) + S ) - N ) ) )
77	45, 76	sylan9eqr 2678	. . . . . 6 |- ( ( ( ph /\ S =/= 0 ) /\ x = ( # ` H ) ) -> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) = ( P ` ( ( ( # ` H ) + S ) - N ) ) )
78	1, 2, 3, 4, 5, 6	crctcshlem2 26710	. . . . . . . . . . . . 13 |- ( ph -> ( # ` H ) = N )
79	78, 24	eqeltrd 2701	. . . . . . . . . . . 12 |- ( ph -> ( # ` H ) e. NN0 )
80	79	nn0cnd 11353	. . . . . . . . . . 11 |- ( ph -> ( # ` H ) e. CC )
81	27	zcnd 11483	. . . . . . . . . . 11 |- ( ph -> S e. CC )
82	24	nn0cnd 11353	. . . . . . . . . . 11 |- ( ph -> N e. CC )
83	80, 81, 82	addsubd 10413	. . . . . . . . . 10 |- ( ph -> ( ( ( # ` H ) + S ) - N ) = ( ( ( # ` H ) - N ) + S ) )
84	78	oveq1d 6665	. . . . . . . . . . . 12 |- ( ph -> ( ( # ` H ) - N ) = ( N - N ) )
85	82	subidd 10380	. . . . . . . . . . . 12 |- ( ph -> ( N - N ) = 0 )
86	84, 85	eqtrd 2656	. . . . . . . . . . 11 |- ( ph -> ( ( # ` H ) - N ) = 0 )
87	86	oveq1d 6665	. . . . . . . . . 10 |- ( ph -> ( ( ( # ` H ) - N ) + S ) = ( 0 + S ) )
88	83, 87	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( ( # ` H ) + S ) - N ) = ( 0 + S ) )
89	88	fveq2d 6195	. . . . . . . 8 |- ( ph -> ( P ` ( ( ( # ` H ) + S ) - N ) ) = ( P ` ( 0 + S ) ) )
90	89	adantr 481	. . . . . . 7 |- ( ( ph /\ S =/= 0 ) -> ( P ` ( ( ( # ` H ) + S ) - N ) ) = ( P ` ( 0 + S ) ) )
91	90	adantr 481	. . . . . 6 |- ( ( ( ph /\ S =/= 0 ) /\ x = ( # ` H ) ) -> ( P ` ( ( ( # ` H ) + S ) - N ) ) = ( P ` ( 0 + S ) ) )
92	77, 91	eqtrd 2656	. . . . 5 |- ( ( ( ph /\ S =/= 0 ) /\ x = ( # ` H ) ) -> if ( x <_ ( N - S ) , ( P ` ( x + S ) ) , ( P ` ( ( x + S ) - N ) ) ) = ( P ` ( 0 + S ) ) )
93	78	adantr 481	. . . . . 6 |- ( ( ph /\ S =/= 0 ) -> ( # ` H ) = N )
94		nn0fz0 12437	. . . . . . . 8 |- ( N e. NN0 <-> N e. ( 0 ... N ) )
95	24, 94	sylib 208	. . . . . . 7 |- ( ph -> N e. ( 0 ... N ) )
96	95	adantr 481	. . . . . 6 |- ( ( ph /\ S =/= 0 ) -> N e. ( 0 ... N ) )
97	93, 96	eqeltrd 2701	. . . . 5 |- ( ( ph /\ S =/= 0 ) -> ( # ` H ) e. ( 0 ... N ) )
98	15, 92, 97, 38	fvmptd 6288	. . . 4 |- ( ( ph /\ S =/= 0 ) -> ( Q ` ( # ` H ) ) = ( P ` ( 0 + S ) ) )
99	39, 98	eqtr4d 2659	. . 3 |- ( ( ph /\ S =/= 0 ) -> ( Q ` 0 ) = ( Q ` ( # ` H ) ) )
100		iscrct 26685	. . 3 |- ( H (Circuits ` G ) Q <-> ( H (Trails ` G ) Q /\ ( Q ` 0 ) = ( Q ` ( # ` H ) ) ) )
101	14, 99, 100	sylanbrc 698	. 2 |- ( ( ph /\ S =/= 0 ) -> H (Circuits ` G ) Q )
102	12, 101	pm2.61dane 2881	1 |- ( ph -> H (Circuits ` G ) Q )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117   cyclShift ccsh 13534  Vtxcvtx 25874  iEdgciedg 25875  Trailsctrls 26587  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-2 11079  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:  eucrctshift  27103