Theorem iseqf1olemnab 10458

Description: Lemma for seq3f1o 10474. (Contributed by Jim Kingdon, 27-Aug-2022.)

Hypotheses

Ref	Expression
iseqf1olemqcl.k	|- ( ph -> K e. ( M ... N ) )
iseqf1olemqcl.j	|- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
iseqf1olemqcl.a	|- ( ph -> A e. ( M ... N ) )
iseqf1olemnab.b	|- ( ph -> B e. ( M ... N ) )
iseqf1olemnab.eq	|- ( ph -> ( Q ` A ) = ( Q ` B ) )
iseqf1olemnab.q	|- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )

Assertion

Ref	Expression
iseqf1olemnab	|- ( ph -> -. ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) )

Distinct variable groups:   u, A   u, B   u, J   u, K   u, M   u, N

Allowed substitution hints:   ph( u)   Q( u)

Proof of Theorem iseqf1olemnab

Step	Hyp	Ref	Expression
1		iseqf1olemnab.eq	. . . 4 |- ( ph -> ( Q ` A ) = ( Q ` B ) )
2	1	adantr 276	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` A ) = ( Q ` B ) )
3		iseqf1olemqcl.k	. . . . . . 7 |- ( ph -> K e. ( M ... N ) )
4		iseqf1olemqcl.j	. . . . . . 7 |- ( ph -> J : ( M ... N ) -1-1-onto-> ( M ... N ) )
5		iseqf1olemqcl.a	. . . . . . 7 |- ( ph -> A e. ( M ... N ) )
6		iseqf1olemnab.q	. . . . . . 7 |- Q = ( u e. ( M ... N ) |-> if ( u e. ( K ... ( `' J ` K ) ) , if ( u = K , K , ( J ` ( u - 1 ) ) ) , ( J ` u ) ) )
7	3, 4, 5, 6	iseqf1olemqval 10457	. . . . . 6 |- ( ph -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )
8	7	adantr 276	. . . . 5 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )
9		simprl 529	. . . . . 6 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> A e. ( K ... ( `' J ` K ) ) )
10	9	iftrued 3539	. . . . 5 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) = if ( A = K , K , ( J ` ( A - 1 ) ) ) )
11	8, 10	eqtrd 2208	. . . 4 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` A ) = if ( A = K , K , ( J ` ( A - 1 ) ) ) )
12		f1ocnvfv2 5769	. . . . . . . 8 |- ( ( J : ( M ... N ) -1-1-onto-> ( M ... N ) /\ K e. ( M ... N ) ) -> ( J ` ( `' J ` K ) ) = K )
13	4, 3, 12	syl2anc 411	. . . . . . 7 |- ( ph -> ( J ` ( `' J ` K ) ) = K )
14	13	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> ( J ` ( `' J ` K ) ) = K )
15		f1ofn 5454	. . . . . . . . 9 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> J Fn ( M ... N ) )
16	4, 15	syl 14	. . . . . . . 8 |- ( ph -> J Fn ( M ... N ) )
17	16	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> J Fn ( M ... N ) )
18		elfzuz 9991	. . . . . . . . . 10 |- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
19		fzss1 10033	. . . . . . . . . 10 |- ( K e. ( ZZ>= ` M ) -> ( K ... ( `' J ` K ) ) C_ ( M ... ( `' J ` K ) ) )
20	3, 18, 19	3syl 17	. . . . . . . . 9 |- ( ph -> ( K ... ( `' J ` K ) ) C_ ( M ... ( `' J ` K ) ) )
21		f1ocnv 5466	. . . . . . . . . . . 12 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
22		f1of 5453	. . . . . . . . . . . 12 |- ( `' J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) --> ( M ... N ) )
23	4, 21, 22	3syl 17	. . . . . . . . . . 11 |- ( ph -> `' J : ( M ... N ) --> ( M ... N ) )
24	23, 3	ffvelcdmd 5644	. . . . . . . . . 10 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
25		elfzuz3 9992	. . . . . . . . . 10 |- ( ( `' J ` K ) e. ( M ... N ) -> N e. ( ZZ>= ` ( `' J ` K ) ) )
26		fzss2 10034	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` ( `' J ` K ) ) -> ( M ... ( `' J ` K ) ) C_ ( M ... N ) )
27	24, 25, 26	3syl 17	. . . . . . . . 9 |- ( ph -> ( M ... ( `' J ` K ) ) C_ ( M ... N ) )
28	20, 27	sstrd 3163	. . . . . . . 8 |- ( ph -> ( K ... ( `' J ` K ) ) C_ ( M ... N ) )
29	28	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> ( K ... ( `' J ` K ) ) C_ ( M ... N ) )
30		elfzubelfz 10006	. . . . . . . . 9 |- ( A e. ( K ... ( `' J ` K ) ) -> ( `' J ` K ) e. ( K ... ( `' J ` K ) ) )
31	30	adantr 276	. . . . . . . 8 |- ( ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> ( `' J ` K ) e. ( K ... ( `' J ` K ) ) )
32	31	ad2antlr 489	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> ( `' J ` K ) e. ( K ... ( `' J ` K ) ) )
33		fnfvima 5742	. . . . . . 7 |- ( ( J Fn ( M ... N ) /\ ( K ... ( `' J ` K ) ) C_ ( M ... N ) /\ ( `' J ` K ) e. ( K ... ( `' J ` K ) ) ) -> ( J ` ( `' J ` K ) ) e. ( J " ( K ... ( `' J ` K ) ) ) )
34	17, 29, 32, 33	syl3anc 1238	. . . . . 6 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> ( J ` ( `' J ` K ) ) e. ( J " ( K ... ( `' J ` K ) ) ) )
35	14, 34	eqeltrrd 2253	. . . . 5 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> K e. ( J " ( K ... ( `' J ` K ) ) ) )
36	16	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> J Fn ( M ... N ) )
37	28	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K ... ( `' J ` K ) ) C_ ( M ... N ) )
38	3	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> K e. ( M ... N ) )
39		elfzelz 9995	. . . . . . . . . 10 |- ( K e. ( M ... N ) -> K e. ZZ )
40	38, 39	syl 14	. . . . . . . . 9 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> K e. ZZ )
41	40	adantr 276	. . . . . . . 8 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> K e. ZZ )
42	24	ad2antrr 488	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( `' J ` K ) e. ( M ... N ) )
43		elfzelz 9995	. . . . . . . . 9 |- ( ( `' J ` K ) e. ( M ... N ) -> ( `' J ` K ) e. ZZ )
44	42, 43	syl 14	. . . . . . . 8 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( `' J ` K ) e. ZZ )
45	5	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> A e. ( M ... N ) )
46		elfzelz 9995	. . . . . . . . . . 11 |- ( A e. ( M ... N ) -> A e. ZZ )
47	45, 46	syl 14	. . . . . . . . . 10 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> A e. ZZ )
48	47	adantr 276	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> A e. ZZ )
49		peano2zm 9264	. . . . . . . . 9 |- ( A e. ZZ -> ( A - 1 ) e. ZZ )
50	48, 49	syl 14	. . . . . . . 8 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) e. ZZ )
51	41, 44, 50	3jca 1177	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K e. ZZ /\ ( `' J ` K ) e. ZZ /\ ( A - 1 ) e. ZZ ) )
52		simpr 110	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> -. A = K )
53		eqcom 2177	. . . . . . . . . . 11 |- ( A = K <-> K = A )
54	52, 53	sylnib 676	. . . . . . . . . 10 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> -. K = A )
55	9	adantr 276	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> A e. ( K ... ( `' J ` K ) ) )
56		elfzle1 9997	. . . . . . . . . . . 12 |- ( A e. ( K ... ( `' J ` K ) ) -> K <_ A )
57	55, 56	syl 14	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> K <_ A )
58		zleloe 9273	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ A e. ZZ ) -> ( K <_ A <-> ( K < A \/ K = A ) ) )
59	41, 48, 58	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K <_ A <-> ( K < A \/ K = A ) ) )
60	57, 59	mpbid 147	. . . . . . . . . 10 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K < A \/ K = A ) )
61	54, 60	ecased 1349	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> K < A )
62		zltlem1 9283	. . . . . . . . . 10 |- ( ( K e. ZZ /\ A e. ZZ ) -> ( K < A <-> K <_ ( A - 1 ) ) )
63	41, 48, 62	syl2anc 411	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K < A <-> K <_ ( A - 1 ) ) )
64	61, 63	mpbid 147	. . . . . . . 8 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> K <_ ( A - 1 ) )
65	50	zred 9348	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) e. RR )
66	48	zred 9348	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> A e. RR )
67	44	zred 9348	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( `' J ` K ) e. RR )
68	66	lem1d 8863	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) <_ A )
69		elfzle2 9998	. . . . . . . . . 10 |- ( A e. ( K ... ( `' J ` K ) ) -> A <_ ( `' J ` K ) )
70	55, 69	syl 14	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> A <_ ( `' J ` K ) )
71	65, 66, 67, 68, 70	letrd 8055	. . . . . . . 8 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) <_ ( `' J ` K ) )
72	64, 71	jca 306	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K <_ ( A - 1 ) /\ ( A - 1 ) <_ ( `' J ` K ) ) )
73		elfz2 9986	. . . . . . 7 |- ( ( A - 1 ) e. ( K ... ( `' J ` K ) ) <-> ( ( K e. ZZ /\ ( `' J ` K ) e. ZZ /\ ( A - 1 ) e. ZZ ) /\ ( K <_ ( A - 1 ) /\ ( A - 1 ) <_ ( `' J ` K ) ) ) )
74	51, 72, 73	sylanbrc 417	. . . . . 6 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) e. ( K ... ( `' J ` K ) ) )
75		fnfvima 5742	. . . . . 6 |- ( ( J Fn ( M ... N ) /\ ( K ... ( `' J ` K ) ) C_ ( M ... N ) /\ ( A - 1 ) e. ( K ... ( `' J ` K ) ) ) -> ( J ` ( A - 1 ) ) e. ( J " ( K ... ( `' J ` K ) ) ) )
76	36, 37, 74, 75	syl3anc 1238	. . . . 5 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( J ` ( A - 1 ) ) e. ( J " ( K ... ( `' J ` K ) ) ) )
77		zdceq 9301	. . . . . 6 |- ( ( A e. ZZ /\ K e. ZZ ) -> DECID A = K )
78	47, 40, 77	syl2anc 411	. . . . 5 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> DECID A = K )
79	35, 76, 78	ifcldadc 3561	. . . 4 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> if ( A = K , K , ( J ` ( A - 1 ) ) ) e. ( J " ( K ... ( `' J ` K ) ) ) )
80	11, 79	eqeltrd 2252	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` A ) e. ( J " ( K ... ( `' J ` K ) ) ) )
81	2, 80	eqeltrrd 2253	. 2 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) )
82		iseqf1olemnab.b	. . . . . 6 |- ( ph -> B e. ( M ... N ) )
83	3, 4, 82, 6	iseqf1olemqval 10457	. . . . 5 |- ( ph -> ( Q ` B ) = if ( B e. ( K ... ( `' J ` K ) ) , if ( B = K , K , ( J ` ( B - 1 ) ) ) , ( J ` B ) ) )
84	83	adantr 276	. . . 4 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` B ) = if ( B e. ( K ... ( `' J ` K ) ) , if ( B = K , K , ( J ` ( B - 1 ) ) ) , ( J ` B ) ) )
85		simprr 531	. . . . 5 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> -. B e. ( K ... ( `' J ` K ) ) )
86	85	iffalsed 3542	. . . 4 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> if ( B e. ( K ... ( `' J ` K ) ) , if ( B = K , K , ( J ` ( B - 1 ) ) ) , ( J ` B ) ) = ( J ` B ) )
87	84, 86	eqtrd 2208	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` B ) = ( J ` B ) )
88		f1of1 5452	. . . . . . 7 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> J : ( M ... N ) -1-1-> ( M ... N ) )
89	4, 88	syl 14	. . . . . 6 |- ( ph -> J : ( M ... N ) -1-1-> ( M ... N ) )
90		f1elima 5764	. . . . . 6 |- ( ( J : ( M ... N ) -1-1-> ( M ... N ) /\ B e. ( M ... N ) /\ ( K ... ( `' J ` K ) ) C_ ( M ... N ) ) -> ( ( J ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) <-> B e. ( K ... ( `' J ` K ) ) ) )
91	89, 82, 28, 90	syl3anc 1238	. . . . 5 |- ( ph -> ( ( J ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) <-> B e. ( K ... ( `' J ` K ) ) ) )
92	91	adantr 276	. . . 4 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( ( J ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) <-> B e. ( K ... ( `' J ` K ) ) ) )
93	85, 92	mtbird 673	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> -. ( J ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) )
94	87, 93	eqneltrd 2271	. 2 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> -. ( Q ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) )
95	81, 94	pm2.65da 661	1 |- ( ph -> -. ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2146   C_ wss 3127   ifcif 3532   class class class wbr 3998   |-> cmpt 4059   `'ccnv 4619   "cima 4623   Fn wfn 5203   -->wf 5204   -1-1->wf1 5205   -1-1-onto->wf1o 5207   ` cfv 5208  (class class class)co 5865   1c1 7787   < clt 7966   <_ cle 7967   - cmin 8102   ZZcz 9226   ZZ>=cuz 9501   ...cfz 9979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-addass 7888  ax-distr 7890  ax-i2m1 7891  ax-0lt1 7892  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897  ax-pre-ltirr 7898  ax-pre-ltwlin 7899  ax-pre-lttrn 7900  ax-pre-ltadd 7902

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-if 3533  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971  df-le 7972  df-sub 8104  df-neg 8105  df-inn 8893  df-n0 9150  df-z 9227  df-uz 9502  df-fz 9980

This theorem is referenced by:  iseqf1olemmo  10462