Theorem pfx2 41412

Description: A prefix of length 2. (Contributed by AV, 15-May-2020.)

Assertion

Ref	Expression
pfx2	|- ( ( W e. Word V /\ 2 <_ ( # ` W ) ) -> ( W prefix 2 ) = <" ( W ` 0 ) ( W ` 1 ) "> )

Proof of Theorem pfx2

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		2nn0 11309	. . . 4 |- 2 e. NN0
2	1	a1i 11	. . 3 |- ( ( W e. Word V /\ 2 <_ ( # ` W ) ) -> 2 e. NN0 )
3		lencl 13324	. . . 4 |- ( W e. Word V -> ( # ` W ) e. NN0 )
4	3	adantr 481	. . 3 |- ( ( W e. Word V /\ 2 <_ ( # ` W ) ) -> ( # ` W ) e. NN0 )
5		simpr 477	. . 3 |- ( ( W e. Word V /\ 2 <_ ( # ` W ) ) -> 2 <_ ( # ` W ) )
6		elfz2nn0 12431	. . 3 |- ( 2 e. ( 0 ... ( # ` W ) ) <-> ( 2 e. NN0 /\ ( # ` W ) e. NN0 /\ 2 <_ ( # ` W ) ) )
7	2, 4, 5, 6	syl3anbrc 1246	. 2 |- ( ( W e. Word V /\ 2 <_ ( # ` W ) ) -> 2 e. ( 0 ... ( # ` W ) ) )
8		pfxlen 41391	. . . 4 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( # ` ( W prefix 2 ) ) = 2 )
9		s2len 13634	. . . . . . 7 |- ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) = 2
10	9	eqcomi 2631	. . . . . 6 |- 2 = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> )
11	10	a1i 11	. . . . 5 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> 2 = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) )
12		simpl 473	. . . . . . . . . 10 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> W e. Word V )
13		simpr 477	. . . . . . . . . 10 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> 2 e. ( 0 ... ( # ` W ) ) )
14		2nn 11185	. . . . . . . . . . . 12 |- 2 e. NN
15		lbfzo0 12507	. . . . . . . . . . . 12 |- ( 0 e. ( 0..^ 2 ) <-> 2 e. NN )
16	14, 15	mpbir 221	. . . . . . . . . . 11 |- 0 e. ( 0..^ 2 )
17	16	a1i 11	. . . . . . . . . 10 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> 0 e. ( 0..^ 2 ) )
18	12, 13, 17	3jca 1242	. . . . . . . . 9 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) /\ 0 e. ( 0..^ 2 ) ) )
19	18	adantr 481	. . . . . . . 8 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) /\ 0 e. ( 0..^ 2 ) ) )
20		pfxfv 41399	. . . . . . . 8 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) /\ 0 e. ( 0..^ 2 ) ) -> ( ( W prefix 2 ) ` 0 ) = ( W ` 0 ) )
21	19, 20	syl 17	. . . . . . 7 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> ( ( W prefix 2 ) ` 0 ) = ( W ` 0 ) )
22		fvex 6201	. . . . . . . 8 |- ( W ` 0 ) e. _V
23		s2fv0 13632	. . . . . . . 8 |- ( ( W ` 0 ) e. _V -> ( <" ( W ` 0 ) ( W ` 1 ) "> ` 0 ) = ( W ` 0 ) )
24	22, 23	ax-mp 5	. . . . . . 7 |- ( <" ( W ` 0 ) ( W ` 1 ) "> ` 0 ) = ( W ` 0 )
25	21, 24	syl6eqr 2674	. . . . . 6 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> ( ( W prefix 2 ) ` 0 ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 0 ) )
26		1nn0 11308	. . . . . . . . . 10 |- 1 e. NN0
27		1lt2 11194	. . . . . . . . . 10 |- 1 < 2
28		elfzo0 12508	. . . . . . . . . 10 |- ( 1 e. ( 0..^ 2 ) <-> ( 1 e. NN0 /\ 2 e. NN /\ 1 < 2 ) )
29	26, 14, 27, 28	mpbir3an 1244	. . . . . . . . 9 |- 1 e. ( 0..^ 2 )
30		pfxfv 41399	. . . . . . . . 9 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) /\ 1 e. ( 0..^ 2 ) ) -> ( ( W prefix 2 ) ` 1 ) = ( W ` 1 ) )
31	29, 30	mp3an3 1413	. . . . . . . 8 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix 2 ) ` 1 ) = ( W ` 1 ) )
32		fvex 6201	. . . . . . . . 9 |- ( W ` 1 ) e. _V
33		s2fv1 13633	. . . . . . . . 9 |- ( ( W ` 1 ) e. _V -> ( <" ( W ` 0 ) ( W ` 1 ) "> ` 1 ) = ( W ` 1 ) )
34	32, 33	ax-mp 5	. . . . . . . 8 |- ( <" ( W ` 0 ) ( W ` 1 ) "> ` 1 ) = ( W ` 1 )
35	31, 34	syl6eqr 2674	. . . . . . 7 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix 2 ) ` 1 ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 1 ) )
36	35	adantr 481	. . . . . 6 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> ( ( W prefix 2 ) ` 1 ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 1 ) )
37		0nn0 11307	. . . . . . . . 9 |- 0 e. NN0
38	37, 26	pm3.2i 471	. . . . . . . 8 |- ( 0 e. NN0 /\ 1 e. NN0 )
39	38	a1i 11	. . . . . . 7 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> ( 0 e. NN0 /\ 1 e. NN0 ) )
40		fveq2 6191	. . . . . . . . 9 |- ( i = 0 -> ( ( W prefix 2 ) ` i ) = ( ( W prefix 2 ) ` 0 ) )
41		fveq2 6191	. . . . . . . . 9 |- ( i = 0 -> ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 0 ) )
42	40, 41	eqeq12d 2637	. . . . . . . 8 |- ( i = 0 -> ( ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) <-> ( ( W prefix 2 ) ` 0 ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 0 ) ) )
43		fveq2 6191	. . . . . . . . 9 |- ( i = 1 -> ( ( W prefix 2 ) ` i ) = ( ( W prefix 2 ) ` 1 ) )
44		fveq2 6191	. . . . . . . . 9 |- ( i = 1 -> ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 1 ) )
45	43, 44	eqeq12d 2637	. . . . . . . 8 |- ( i = 1 -> ( ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) <-> ( ( W prefix 2 ) ` 1 ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 1 ) ) )
46	42, 45	ralprg 4234	. . . . . . 7 |- ( ( 0 e. NN0 /\ 1 e. NN0 ) -> ( A. i e. { 0 , 1 } ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) <-> ( ( ( W prefix 2 ) ` 0 ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 0 ) /\ ( ( W prefix 2 ) ` 1 ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 1 ) ) ) )
47	39, 46	syl 17	. . . . . 6 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> ( A. i e. { 0 , 1 } ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) <-> ( ( ( W prefix 2 ) ` 0 ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 0 ) /\ ( ( W prefix 2 ) ` 1 ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` 1 ) ) ) )
48	25, 36, 47	mpbir2and 957	. . . . 5 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> A. i e. { 0 , 1 } ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) )
49		eqeq1 2626	. . . . . . 7 |- ( ( # ` ( W prefix 2 ) ) = 2 -> ( ( # ` ( W prefix 2 ) ) = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) <-> 2 = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) ) )
50		oveq2 6658	. . . . . . . . 9 |- ( ( # ` ( W prefix 2 ) ) = 2 -> ( 0..^ ( # ` ( W prefix 2 ) ) ) = ( 0..^ 2 ) )
51		fzo0to2pr 12553	. . . . . . . . 9 |- ( 0..^ 2 ) = { 0 , 1 }
52	50, 51	syl6eq 2672	. . . . . . . 8 |- ( ( # ` ( W prefix 2 ) ) = 2 -> ( 0..^ ( # ` ( W prefix 2 ) ) ) = { 0 , 1 } )
53	52	raleqdv 3144	. . . . . . 7 |- ( ( # ` ( W prefix 2 ) ) = 2 -> ( A. i e. ( 0..^ ( # ` ( W prefix 2 ) ) ) ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) <-> A. i e. { 0 , 1 } ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) ) )
54	49, 53	anbi12d 747	. . . . . 6 |- ( ( # ` ( W prefix 2 ) ) = 2 -> ( ( ( # ` ( W prefix 2 ) ) = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) /\ A. i e. ( 0..^ ( # ` ( W prefix 2 ) ) ) ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) ) <-> ( 2 = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) /\ A. i e. { 0 , 1 } ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) ) ) )
55	54	adantl 482	. . . . 5 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> ( ( ( # ` ( W prefix 2 ) ) = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) /\ A. i e. ( 0..^ ( # ` ( W prefix 2 ) ) ) ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) ) <-> ( 2 = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) /\ A. i e. { 0 , 1 } ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) ) ) )
56	11, 48, 55	mpbir2and 957	. . . 4 |- ( ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) /\ ( # ` ( W prefix 2 ) ) = 2 ) -> ( ( # ` ( W prefix 2 ) ) = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) /\ A. i e. ( 0..^ ( # ` ( W prefix 2 ) ) ) ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) ) )
57	8, 56	mpdan 702	. . 3 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( ( # ` ( W prefix 2 ) ) = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) /\ A. i e. ( 0..^ ( # ` ( W prefix 2 ) ) ) ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) ) )
58		pfxcl 41386	. . . . 5 |- ( W e. Word V -> ( W prefix 2 ) e. Word V )
59	58	adantr 481	. . . 4 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( W prefix 2 ) e. Word V )
60		simp2 1062	. . . . . . . . . 10 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 /\ 2 <_ ( # ` W ) ) -> ( # ` W ) e. NN0 )
61		1red 10055	. . . . . . . . . . . . . . 15 |- ( ( # ` W ) e. NN0 -> 1 e. RR )
62		2re 11090	. . . . . . . . . . . . . . . 16 |- 2 e. RR
63	62	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( # ` W ) e. NN0 -> 2 e. RR )
64		nn0re 11301	. . . . . . . . . . . . . . 15 |- ( ( # ` W ) e. NN0 -> ( # ` W ) e. RR )
65	61, 63, 64	3jca 1242	. . . . . . . . . . . . . 14 |- ( ( # ` W ) e. NN0 -> ( 1 e. RR /\ 2 e. RR /\ ( # ` W ) e. RR ) )
66		ltleletr 10130	. . . . . . . . . . . . . 14 |- ( ( 1 e. RR /\ 2 e. RR /\ ( # ` W ) e. RR ) -> ( ( 1 < 2 /\ 2 <_ ( # ` W ) ) -> 1 <_ ( # ` W ) ) )
67	65, 66	syl 17	. . . . . . . . . . . . 13 |- ( ( # ` W ) e. NN0 -> ( ( 1 < 2 /\ 2 <_ ( # ` W ) ) -> 1 <_ ( # ` W ) ) )
68	27, 67	mpani 712	. . . . . . . . . . . 12 |- ( ( # ` W ) e. NN0 -> ( 2 <_ ( # ` W ) -> 1 <_ ( # ` W ) ) )
69	68	imp 445	. . . . . . . . . . 11 |- ( ( ( # ` W ) e. NN0 /\ 2 <_ ( # ` W ) ) -> 1 <_ ( # ` W ) )
70	69	3adant1 1079	. . . . . . . . . 10 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 /\ 2 <_ ( # ` W ) ) -> 1 <_ ( # ` W ) )
71	60, 70	jca 554	. . . . . . . . 9 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 /\ 2 <_ ( # ` W ) ) -> ( ( # ` W ) e. NN0 /\ 1 <_ ( # ` W ) ) )
72		elnnnn0c 11338	. . . . . . . . 9 |- ( ( # ` W ) e. NN <-> ( ( # ` W ) e. NN0 /\ 1 <_ ( # ` W ) ) )
73	71, 72	sylibr 224	. . . . . . . 8 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 /\ 2 <_ ( # ` W ) ) -> ( # ` W ) e. NN )
74	6, 73	sylbi 207	. . . . . . 7 |- ( 2 e. ( 0 ... ( # ` W ) ) -> ( # ` W ) e. NN )
75		lbfzo0 12507	. . . . . . 7 |- ( 0 e. ( 0..^ ( # ` W ) ) <-> ( # ` W ) e. NN )
76	74, 75	sylibr 224	. . . . . 6 |- ( 2 e. ( 0 ... ( # ` W ) ) -> 0 e. ( 0..^ ( # ` W ) ) )
77		wrdsymbcl 13318	. . . . . 6 |- ( ( W e. Word V /\ 0 e. ( 0..^ ( # ` W ) ) ) -> ( W ` 0 ) e. V )
78	76, 77	sylan2 491	. . . . 5 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( W ` 0 ) e. V )
79	26	a1i 11	. . . . . . 7 |- ( 2 e. ( 0 ... ( # ` W ) ) -> 1 e. NN0 )
80	65	adantl 482	. . . . . . . . . . 11 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) -> ( 1 e. RR /\ 2 e. RR /\ ( # ` W ) e. RR ) )
81		ltletr 10129	. . . . . . . . . . 11 |- ( ( 1 e. RR /\ 2 e. RR /\ ( # ` W ) e. RR ) -> ( ( 1 < 2 /\ 2 <_ ( # ` W ) ) -> 1 < ( # ` W ) ) )
82	80, 81	syl 17	. . . . . . . . . 10 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) -> ( ( 1 < 2 /\ 2 <_ ( # ` W ) ) -> 1 < ( # ` W ) ) )
83	27, 82	mpani 712	. . . . . . . . 9 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 ) -> ( 2 <_ ( # ` W ) -> 1 < ( # ` W ) ) )
84	83	3impia 1261	. . . . . . . 8 |- ( ( 2 e. NN0 /\ ( # ` W ) e. NN0 /\ 2 <_ ( # ` W ) ) -> 1 < ( # ` W ) )
85	6, 84	sylbi 207	. . . . . . 7 |- ( 2 e. ( 0 ... ( # ` W ) ) -> 1 < ( # ` W ) )
86		elfzo0 12508	. . . . . . 7 |- ( 1 e. ( 0..^ ( # ` W ) ) <-> ( 1 e. NN0 /\ ( # ` W ) e. NN /\ 1 < ( # ` W ) ) )
87	79, 74, 85, 86	syl3anbrc 1246	. . . . . 6 |- ( 2 e. ( 0 ... ( # ` W ) ) -> 1 e. ( 0..^ ( # ` W ) ) )
88		wrdsymbcl 13318	. . . . . 6 |- ( ( W e. Word V /\ 1 e. ( 0..^ ( # ` W ) ) ) -> ( W ` 1 ) e. V )
89	87, 88	sylan2 491	. . . . 5 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( W ` 1 ) e. V )
90	78, 89	s2cld 13616	. . . 4 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> <" ( W ` 0 ) ( W ` 1 ) "> e. Word V )
91		eqwrd 13346	. . . 4 |- ( ( ( W prefix 2 ) e. Word V /\ <" ( W ` 0 ) ( W ` 1 ) "> e. Word V ) -> ( ( W prefix 2 ) = <" ( W ` 0 ) ( W ` 1 ) "> <-> ( ( # ` ( W prefix 2 ) ) = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) /\ A. i e. ( 0..^ ( # ` ( W prefix 2 ) ) ) ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) ) ) )
92	59, 90, 91	syl2anc 693	. . 3 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( ( W prefix 2 ) = <" ( W ` 0 ) ( W ` 1 ) "> <-> ( ( # ` ( W prefix 2 ) ) = ( # ` <" ( W ` 0 ) ( W ` 1 ) "> ) /\ A. i e. ( 0..^ ( # ` ( W prefix 2 ) ) ) ( ( W prefix 2 ) ` i ) = ( <" ( W ` 0 ) ( W ` 1 ) "> ` i ) ) ) )
93	57, 92	mpbird 247	. 2 |- ( ( W e. Word V /\ 2 e. ( 0 ... ( # ` W ) ) ) -> ( W prefix 2 ) = <" ( W ` 0 ) ( W ` 1 ) "> )
94	7, 93	syldan 487	1 |- ( ( W e. Word V /\ 2 <_ ( # ` W ) ) -> ( W prefix 2 ) = <" ( W ` 0 ) ( W ` 1 ) "> )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   {cpr 4179   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   NNcn 11020   2c2 11070   NN0cn0 11292   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   <"cs2 13586   prefix cpfx 41381

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  df-s2 13593  df-pfx 41382

This theorem is referenced by: (None)