Theorem numclwlk1lem2foalem 27222

Description: Lemma for numclwlk1lem2foa 27224. (Contributed by AV, 29-May-2021.)

Assertion

Ref	Expression
numclwlk1lem2foalem	|- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = W /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = Y /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) )

Proof of Theorem numclwlk1lem2foalem

Step	Hyp	Ref	Expression
1		simpl 473	. . . . . . . 8 |- ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) -> W e. Word V )
2		s1cl 13382	. . . . . . . 8 |- ( X e. V -> <" X "> e. Word V )
3		s1cl 13382	. . . . . . . 8 |- ( Y e. V -> <" Y "> e. Word V )
4	1, 2, 3	3anim123i 1247	. . . . . . 7 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ X e. V /\ Y e. V ) -> ( W e. Word V /\ <" X "> e. Word V /\ <" Y "> e. Word V ) )
5	4	3expb 1266	. . . . . 6 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> ( W e. Word V /\ <" X "> e. Word V /\ <" Y "> e. Word V ) )
6		ccatass 13371	. . . . . 6 |- ( ( W e. Word V /\ <" X "> e. Word V /\ <" Y "> e. Word V ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ ( <" X "> ++ <" Y "> ) ) )
7	5, 6	syl 17	. . . . 5 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( W ++ <" X "> ) ++ <" Y "> ) = ( W ++ ( <" X "> ++ <" Y "> ) ) )
8	7	oveq1d 6665	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = ( ( W ++ ( <" X "> ++ <" Y "> ) ) substr <. 0 , ( N - 2 ) >. ) )
9	1	adantr 481	. . . . 5 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> W e. Word V )
10		ccat2s1cl 13397	. . . . . 6 |- ( ( X e. V /\ Y e. V ) -> ( <" X "> ++ <" Y "> ) e. Word V )
11	10	adantl 482	. . . . 5 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> ( <" X "> ++ <" Y "> ) e. Word V )
12		simpr 477	. . . . . . 7 |- ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) -> ( # ` W ) = ( N - 2 ) )
13	12	eqcomd 2628	. . . . . 6 |- ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) -> ( N - 2 ) = ( # ` W ) )
14	13	adantr 481	. . . . 5 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> ( N - 2 ) = ( # ` W ) )
15		swrdccatid 13497	. . . . 5 |- ( ( W e. Word V /\ ( <" X "> ++ <" Y "> ) e. Word V /\ ( N - 2 ) = ( # ` W ) ) -> ( ( W ++ ( <" X "> ++ <" Y "> ) ) substr <. 0 , ( N - 2 ) >. ) = W )
16	9, 11, 14, 15	syl3anc 1326	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( W ++ ( <" X "> ++ <" Y "> ) ) substr <. 0 , ( N - 2 ) >. ) = W )
17	8, 16	eqtrd 2656	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = W )
18	17	3adant3 1081	. 2 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = W )
19		1e2m1 11136	. . . . . . 7 |- 1 = ( 2 - 1 )
20	19	oveq2i 6661	. . . . . 6 |- ( N - 1 ) = ( N - ( 2 - 1 ) )
21		eluzelcn 11699	. . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> N e. CC )
22		2cnd 11093	. . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> 2 e. CC )
23		1cnd 10056	. . . . . . 7 |- ( N e. ( ZZ>= ` 3 ) -> 1 e. CC )
24	21, 22, 23	subsubd 10420	. . . . . 6 |- ( N e. ( ZZ>= ` 3 ) -> ( N - ( 2 - 1 ) ) = ( ( N - 2 ) + 1 ) )
25	20, 24	syl5eq 2668	. . . . 5 |- ( N e. ( ZZ>= ` 3 ) -> ( N - 1 ) = ( ( N - 2 ) + 1 ) )
26	25	3ad2ant3 1084	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) -> ( N - 1 ) = ( ( N - 2 ) + 1 ) )
27	26	fveq2d 6195	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( N - 2 ) + 1 ) ) )
28		ccatw2s1p2 13414	. . . 4 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( N - 2 ) + 1 ) ) = Y )
29	28	3adant3 1081	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( ( N - 2 ) + 1 ) ) = Y )
30	27, 29	eqtrd 2656	. 2 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = Y )
31		ccatw2s1p1 13413	. . 3 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X )
32	31	3adant3 1081	. 2 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X )
33	18, 30, 32	3jca 1242	1 |- ( ( ( W e. Word V /\ ( # ` W ) = ( N - 2 ) ) /\ ( X e. V /\ Y e. V ) /\ N e. ( ZZ>= ` 3 ) ) -> ( ( ( ( W ++ <" X "> ) ++ <" Y "> ) substr <. 0 , ( N - 2 ) >. ) = W /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 1 ) ) = Y /\ ( ( ( W ++ <" X "> ) ++ <" Y "> ) ` ( N - 2 ) ) = X ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   <.cop 4183   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   - cmin 10266   2c2 11070   3c3 11071   ZZ>=cuz 11687   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295

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-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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303

This theorem is referenced by:  numclwlk1lem2foa  27224