Theorem iseqf1olemnab 10423

Description: Lemma for seq3f1o 10439. (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 274	. . 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 10422	. . . . . 6 |- ( ph -> ( Q ` A ) = if ( A e. ( K ... ( `' J ` K ) ) , if ( A = K , K , ( J ` ( A - 1 ) ) ) , ( J ` A ) ) )
8	7	adantr 274	. . . . 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 521	. . . . . 6 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> A e. ( K ... ( `' J ` K ) ) )
10	9	iftrued 3527	. . . . 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 2198	. . . 4 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` A ) = if ( A = K , K , ( J ` ( A - 1 ) ) ) )
12		f1ocnvfv2 5746	. . . . . . . 8 |- ( ( J : ( M ... N ) -1-1-onto-> ( M ... N ) /\ K e. ( M ... N ) ) -> ( J ` ( `' J ` K ) ) = K )
13	4, 3, 12	syl2anc 409	. . . . . . 7 |- ( ph -> ( J ` ( `' J ` K ) ) = K )
14	13	ad2antrr 480	. . . . . 6 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> ( J ` ( `' J ` K ) ) = K )
15		f1ofn 5433	. . . . . . . . 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 480	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> J Fn ( M ... N ) )
18		elfzuz 9956	. . . . . . . . . 10 |- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
19		fzss1 9998	. . . . . . . . . 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 5445	. . . . . . . . . . . 12 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
22		f1of 5432	. . . . . . . . . . . 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	ffvelrnd 5621	. . . . . . . . . 10 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
25		elfzuz3 9957	. . . . . . . . . 10 |- ( ( `' J ` K ) e. ( M ... N ) -> N e. ( ZZ>= ` ( `' J ` K ) ) )
26		fzss2 9999	. . . . . . . . . 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 3152	. . . . . . . 8 |- ( ph -> ( K ... ( `' J ` K ) ) C_ ( M ... N ) )
29	28	ad2antrr 480	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> ( K ... ( `' J ` K ) ) C_ ( M ... N ) )
30		elfzubelfz 9971	. . . . . . . . 9 |- ( A e. ( K ... ( `' J ` K ) ) -> ( `' J ` K ) e. ( K ... ( `' J ` K ) ) )
31	30	adantr 274	. . . . . . . 8 |- ( ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) -> ( `' J ` K ) e. ( K ... ( `' J ` K ) ) )
32	31	ad2antlr 481	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> ( `' J ` K ) e. ( K ... ( `' J ` K ) ) )
33		fnfvima 5719	. . . . . . 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 1228	. . . . . 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 2244	. . . . 5 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ A = K ) -> K e. ( J " ( K ... ( `' J ` K ) ) ) )
36	16	ad2antrr 480	. . . . . 6 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> J Fn ( M ... N ) )
37	28	ad2antrr 480	. . . . . 6 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K ... ( `' J ` K ) ) C_ ( M ... N ) )
38	3	adantr 274	. . . . . . . . . 10 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> K e. ( M ... N ) )
39		elfzelz 9960	. . . . . . . . . 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 274	. . . . . . . 8 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> K e. ZZ )
42	24	ad2antrr 480	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( `' J ` K ) e. ( M ... N ) )
43		elfzelz 9960	. . . . . . . . 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 274	. . . . . . . . . . 11 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> A e. ( M ... N ) )
46		elfzelz 9960	. . . . . . . . . . 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 274	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> A e. ZZ )
49		peano2zm 9229	. . . . . . . . 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 1167	. . . . . . 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 109	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> -. A = K )
53		eqcom 2167	. . . . . . . . . . 11 |- ( A = K <-> K = A )
54	52, 53	sylnib 666	. . . . . . . . . 10 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> -. K = A )
55	9	adantr 274	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> A e. ( K ... ( `' J ` K ) ) )
56		elfzle1 9962	. . . . . . . . . . . 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 9238	. . . . . . . . . . . 12 |- ( ( K e. ZZ /\ A e. ZZ ) -> ( K <_ A <-> ( K < A \/ K = A ) ) )
59	41, 48, 58	syl2anc 409	. . . . . . . . . . 11 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K <_ A <-> ( K < A \/ K = A ) ) )
60	57, 59	mpbid 146	. . . . . . . . . 10 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K < A \/ K = A ) )
61	54, 60	ecased 1339	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> K < A )
62		zltlem1 9248	. . . . . . . . . 10 |- ( ( K e. ZZ /\ A e. ZZ ) -> ( K < A <-> K <_ ( A - 1 ) ) )
63	41, 48, 62	syl2anc 409	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K < A <-> K <_ ( A - 1 ) ) )
64	61, 63	mpbid 146	. . . . . . . 8 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> K <_ ( A - 1 ) )
65	50	zred 9313	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) e. RR )
66	48	zred 9313	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> A e. RR )
67	44	zred 9313	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( `' J ` K ) e. RR )
68	66	lem1d 8828	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) <_ A )
69		elfzle2 9963	. . . . . . . . . 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 8022	. . . . . . . 8 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) <_ ( `' J ` K ) )
72	64, 71	jca 304	. . . . . . 7 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( K <_ ( A - 1 ) /\ ( A - 1 ) <_ ( `' J ` K ) ) )
73		elfz2 9951	. . . . . . 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 414	. . . . . 6 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) e. ( K ... ( `' J ` K ) ) )
75		fnfvima 5719	. . . . . 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 1228	. . . . 5 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( J ` ( A - 1 ) ) e. ( J " ( K ... ( `' J ` K ) ) ) )
77		zdceq 9266	. . . . . 6 |- ( ( A e. ZZ /\ K e. ZZ ) -> DECID A = K )
78	47, 40, 77	syl2anc 409	. . . . 5 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> DECID A = K )
79	35, 76, 78	ifcldadc 3549	. . . 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 2243	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` A ) e. ( J " ( K ... ( `' J ` K ) ) ) )
81	2, 80	eqeltrrd 2244	. 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 10422	. . . . 5 |- ( ph -> ( Q ` B ) = if ( B e. ( K ... ( `' J ` K ) ) , if ( B = K , K , ( J ` ( B - 1 ) ) ) , ( J ` B ) ) )
84	83	adantr 274	. . . 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 522	. . . . 5 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> -. B e. ( K ... ( `' J ` K ) ) )
86	85	iffalsed 3530	. . . 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 2198	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` B ) = ( J ` B ) )
88		f1of1 5431	. . . . . . 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 5741	. . . . . 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 1228	. . . . 5 |- ( ph -> ( ( J ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) <-> B e. ( K ... ( `' J ` K ) ) ) )
92	91	adantr 274	. . . 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 663	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> -. ( J ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) )
94	87, 93	eqneltrd 2262	. 2 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> -. ( Q ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) )
95	81, 94	pm2.65da 651	1 |- ( ph -> -. ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698  DECID wdc 824   /\ w3a 968   = wceq 1343   e. wcel 2136   C_ wss 3116   ifcif 3520   class class class wbr 3982   |-> cmpt 4043   `'ccnv 4603   "cima 4607   Fn wfn 5183   -->wf 5184   -1-1->wf1 5185   -1-1-onto->wf1o 5187   ` cfv 5188  (class class class)co 5842   1c1 7754   < clt 7933   <_ cle 7934   - cmin 8069   ZZcz 9191   ZZ>=cuz 9466   ...cfz 9944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945

This theorem is referenced by:  iseqf1olemmo  10427