Theorem 2swrd2eqwrdeq 13696

Description: Two words of length at least 2 are equal if and only if they have the same prefix and the same two single symbols suffix. (Contributed by AV, 24-Sep-2018.) (Revised by Mario Carneiro/AV, 23-Oct-2018.)

Assertion

Ref	Expression
2swrd2eqwrdeq	|- ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) -> ( W = U <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) ) )

Proof of Theorem 2swrd2eqwrdeq

Step	Hyp	Ref	Expression
1		lencl 13324	. . . . 5 |- ( W e. Word V -> ( # ` W ) e. NN0 )
2		1z 11407	. . . . . . . . . 10 |- 1 e. ZZ
3		nn0z 11400	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. ZZ )
4		zltp1le 11427	. . . . . . . . . 10 |- ( ( 1 e. ZZ /\ ( # ` W ) e. ZZ ) -> ( 1 < ( # ` W ) <-> ( 1 + 1 ) <_ ( # ` W ) ) )
5	2, 3, 4	sylancr 695	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) <-> ( 1 + 1 ) <_ ( # ` W ) ) )
6		1p1e2 11134	. . . . . . . . . . . 12 |- ( 1 + 1 ) = 2
7	6	a1i 11	. . . . . . . . . . 11 |- ( ( # ` W ) e. NN0 -> ( 1 + 1 ) = 2 )
8	7	breq1d 4663	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( ( 1 + 1 ) <_ ( # ` W ) <-> 2 <_ ( # ` W ) ) )
9	8	biimpd 219	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( ( 1 + 1 ) <_ ( # ` W ) -> 2 <_ ( # ` W ) ) )
10	5, 9	sylbid 230	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) -> 2 <_ ( # ` W ) ) )
11	10	imp 445	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> 2 <_ ( # ` W ) )
12		2nn0 11309	. . . . . . . 8 |- 2 e. NN0
13		simpl 473	. . . . . . . 8 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( # ` W ) e. NN0 )
14		nn0sub 11343	. . . . . . . 8 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) -> ( 2 <_ ( # ` W ) <-> ( ( # ` W ) - 2 ) e. NN0 ) )
15	12, 13, 14	sylancr 695	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( 2 <_ ( # ` W ) <-> ( ( # ` W ) - 2 ) e. NN0 ) )
16	11, 15	mpbid 222	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. NN0 )
17	3	adantr 481	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( # ` W ) e. ZZ )
18		0red 10041	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> 0 e. RR )
19		1red 10055	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> 1 e. RR )
20		nn0re 11301	. . . . . . . . . 10 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. RR )
21	18, 19, 20	3jca 1242	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) )
22		0lt1 10550	. . . . . . . . 9 |- 0 < 1
23		lttr 10114	. . . . . . . . . 10 |- ( ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) -> ( ( 0 < 1 /\ 1 < ( # ` W ) ) -> 0 < ( # ` W ) ) )
24	23	expd 452	. . . . . . . . 9 |- ( ( 0 e. RR /\ 1 e. RR /\ ( # ` W ) e. RR ) -> ( 0 < 1 -> ( 1 < ( # ` W ) -> 0 < ( # ` W ) ) ) )
25	21, 22, 24	mpisyl 21	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 1 < ( # ` W ) -> 0 < ( # ` W ) ) )
26	25	imp 445	. . . . . . 7 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> 0 < ( # ` W ) )
27		elnnz 11387	. . . . . . 7 |- ( ( # ` W ) e. NN <-> ( ( # ` W ) e. ZZ /\ 0 < ( # ` W ) ) )
28	17, 26, 27	sylanbrc 698	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( # ` W ) e. NN )
29		2pos 11112	. . . . . . . 8 |- 0 < 2
30		2re 11090	. . . . . . . . . 10 |- 2 e. RR
31	30	a1i 11	. . . . . . . . 9 |- ( ( # ` W ) e. NN0 -> 2 e. RR )
32	31, 20	ltsubposd 10613	. . . . . . . 8 |- ( ( # ` W ) e. NN0 -> ( 0 < 2 <-> ( ( # ` W ) - 2 ) < ( # ` W ) ) )
33	29, 32	mpbii 223	. . . . . . 7 |- ( ( # ` W ) e. NN0 -> ( ( # ` W ) - 2 ) < ( # ` W ) )
34	33	adantr 481	. . . . . 6 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) < ( # ` W ) )
35		elfzo0 12508	. . . . . 6 |- ( ( ( # ` W ) - 2 ) e. ( 0..^ ( # ` W ) ) <-> ( ( ( # ` W ) - 2 ) e. NN0 /\ ( # ` W ) e. NN /\ ( ( # ` W ) - 2 ) < ( # ` W ) ) )
36	16, 28, 34, 35	syl3anbrc 1246	. . . . 5 |- ( ( ( # ` W ) e. NN0 /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. ( 0..^ ( # ` W ) ) )
37	1, 36	sylan 488	. . . 4 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. ( 0..^ ( # ` W ) ) )
38	37	3adant2 1080	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) -> ( ( # ` W ) - 2 ) e. ( 0..^ ( # ` W ) ) )
39		2swrdeqwrdeq 13453	. . 3 |- ( ( W e. Word V /\ U e. Word V /\ ( ( # ` W ) - 2 ) e. ( 0..^ ( # ` W ) ) ) -> ( W = U <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) ) ) ) )
40	38, 39	syld3an3 1371	. 2 |- ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) -> ( W = U <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) ) ) ) )
41		swrd2lsw 13695	. . . . . . . . 9 |- ( ( W e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> )
42	41	3adant2 1080	. . . . . . . 8 |- ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> )
43	42	adantr 481	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> )
44		breq2 4657	. . . . . . . . . . 11 |- ( ( # ` W ) = ( # ` U ) -> ( 1 < ( # ` W ) <-> 1 < ( # ` U ) ) )
45	44	3anbi3d 1405	. . . . . . . . . 10 |- ( ( # ` W ) = ( # ` U ) -> ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) <-> ( W e. Word V /\ U e. Word V /\ 1 < ( # ` U ) ) ) )
46		swrd2lsw 13695	. . . . . . . . . . 11 |- ( ( U e. Word V /\ 1 < ( # ` U ) ) -> ( U substr <. ( ( # ` U ) - 2 ) , ( # ` U ) >. ) = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> )
47	46	3adant1 1079	. . . . . . . . . 10 |- ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` U ) ) -> ( U substr <. ( ( # ` U ) - 2 ) , ( # ` U ) >. ) = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> )
48	45, 47	syl6bi 243	. . . . . . . . 9 |- ( ( # ` W ) = ( # ` U ) -> ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) -> ( U substr <. ( ( # ` U ) - 2 ) , ( # ` U ) >. ) = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> ) )
49	48	impcom 446	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( U substr <. ( ( # ` U ) - 2 ) , ( # ` U ) >. ) = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> )
50		oveq1 6657	. . . . . . . . . . . 12 |- ( ( # ` W ) = ( # ` U ) -> ( ( # ` W ) - 2 ) = ( ( # ` U ) - 2 ) )
51		id 22	. . . . . . . . . . . 12 |- ( ( # ` W ) = ( # ` U ) -> ( # ` W ) = ( # ` U ) )
52	50, 51	opeq12d 4410	. . . . . . . . . . 11 |- ( ( # ` W ) = ( # ` U ) -> <. ( ( # ` W ) - 2 ) , ( # ` W ) >. = <. ( ( # ` U ) - 2 ) , ( # ` U ) >. )
53	52	oveq2d 6666	. . . . . . . . . 10 |- ( ( # ` W ) = ( # ` U ) -> ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` U ) - 2 ) , ( # ` U ) >. ) )
54	53	eqeq1d 2624	. . . . . . . . 9 |- ( ( # ` W ) = ( # ` U ) -> ( ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> <-> ( U substr <. ( ( # ` U ) - 2 ) , ( # ` U ) >. ) = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> ) )
55	54	adantl 482	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> <-> ( U substr <. ( ( # ` U ) - 2 ) , ( # ` U ) >. ) = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> ) )
56	49, 55	mpbird 247	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> )
57	43, 56	eqeq12d 2637	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) <-> <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> ) )
58		fvexd 6203	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( W ` ( ( # ` W ) - 2 ) ) e. _V )
59		fvexd 6203	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( lastS ` W ) e. _V )
60		fvexd 6203	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( U ` ( ( # ` U ) - 2 ) ) e. _V )
61		fvexd 6203	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( lastS ` U ) e. _V )
62		s2eq2s1eq 13681	. . . . . . 7 |- ( ( ( ( W ` ( ( # ` W ) - 2 ) ) e. _V /\ ( lastS ` W ) e. _V ) /\ ( ( U ` ( ( # ` U ) - 2 ) ) e. _V /\ ( lastS ` U ) e. _V ) ) -> ( <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> <-> ( <" ( W ` ( ( # ` W ) - 2 ) ) "> = <" ( U ` ( ( # ` U ) - 2 ) ) "> /\ <" ( lastS ` W ) "> = <" ( lastS ` U ) "> ) ) )
63	58, 59, 60, 61, 62	syl22anc 1327	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( <" ( W ` ( ( # ` W ) - 2 ) ) ( lastS ` W ) "> = <" ( U ` ( ( # ` U ) - 2 ) ) ( lastS ` U ) "> <-> ( <" ( W ` ( ( # ` W ) - 2 ) ) "> = <" ( U ` ( ( # ` U ) - 2 ) ) "> /\ <" ( lastS ` W ) "> = <" ( lastS ` U ) "> ) ) )
64		fvex 6201	. . . . . . . . 9 |- ( W ` ( ( # ` W ) - 2 ) ) e. _V
65		s111 13395	. . . . . . . . 9 |- ( ( ( W ` ( ( # ` W ) - 2 ) ) e. _V /\ ( U ` ( ( # ` U ) - 2 ) ) e. _V ) -> ( <" ( W ` ( ( # ` W ) - 2 ) ) "> = <" ( U ` ( ( # ` U ) - 2 ) ) "> <-> ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` U ) - 2 ) ) ) )
66	64, 60, 65	sylancr 695	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( <" ( W ` ( ( # ` W ) - 2 ) ) "> = <" ( U ` ( ( # ` U ) - 2 ) ) "> <-> ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` U ) - 2 ) ) ) )
67		oveq1 6657	. . . . . . . . . . . 12 |- ( ( # ` U ) = ( # ` W ) -> ( ( # ` U ) - 2 ) = ( ( # ` W ) - 2 ) )
68	67	fveq2d 6195	. . . . . . . . . . 11 |- ( ( # ` U ) = ( # ` W ) -> ( U ` ( ( # ` U ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) )
69	68	eqcoms 2630	. . . . . . . . . 10 |- ( ( # ` W ) = ( # ` U ) -> ( U ` ( ( # ` U ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) )
70	69	adantl 482	. . . . . . . . 9 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( U ` ( ( # ` U ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) )
71	70	eqeq2d 2632	. . . . . . . 8 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` U ) - 2 ) ) <-> ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) ) )
72	66, 71	bitrd 268	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( <" ( W ` ( ( # ` W ) - 2 ) ) "> = <" ( U ` ( ( # ` U ) - 2 ) ) "> <-> ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) ) )
73		fvex 6201	. . . . . . . 8 |- ( lastS ` W ) e. _V
74		s111 13395	. . . . . . . 8 |- ( ( ( lastS ` W ) e. _V /\ ( lastS ` U ) e. _V ) -> ( <" ( lastS ` W ) "> = <" ( lastS ` U ) "> <-> ( lastS ` W ) = ( lastS ` U ) ) )
75	73, 61, 74	sylancr 695	. . . . . . 7 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( <" ( lastS ` W ) "> = <" ( lastS ` U ) "> <-> ( lastS ` W ) = ( lastS ` U ) ) )
76	72, 75	anbi12d 747	. . . . . 6 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( ( <" ( W ` ( ( # ` W ) - 2 ) ) "> = <" ( U ` ( ( # ` U ) - 2 ) ) "> /\ <" ( lastS ` W ) "> = <" ( lastS ` U ) "> ) <-> ( ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) )
77	57, 63, 76	3bitrd 294	. . . . 5 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) <-> ( ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) )
78	77	anbi2d 740	. . . 4 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) ) <-> ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) ) )
79		3anass 1042	. . . 4 |- ( ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) <-> ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) )
80	78, 79	syl6bbr 278	. . 3 |- ( ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) /\ ( # ` W ) = ( # ` U ) ) -> ( ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) ) <-> ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) )
81	80	pm5.32da 673	. 2 |- ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) -> ( ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) = ( U substr <. ( ( # ` W ) - 2 ) , ( # ` W ) >. ) ) ) <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) ) )
82	40, 81	bitrd 268	1 |- ( ( W e. Word V /\ U e. Word V /\ 1 < ( # ` W ) ) -> ( W = U <-> ( ( # ` W ) = ( # ` U ) /\ ( ( W substr <. 0 , ( ( # ` W ) - 2 ) >. ) = ( U substr <. 0 , ( ( # ` W ) - 2 ) >. ) /\ ( W ` ( ( # ` W ) - 2 ) ) = ( U ` ( ( # ` W ) - 2 ) ) /\ ( lastS ` W ) = ( lastS ` U ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   2c2 11070   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   #chash 13117  Word cword 13291   lastS clsw 13292   <"cs1 13294   substr csubstr 13295   <"cs2 13586

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-fal 1489  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-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-s2 13593

This theorem is referenced by:  numclwlk1lem2f1  27227