Theorem iseqf1olemnab 10527

Description: Lemma for seq3f1o 10543. (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 10526	. . . . . 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 3556	. . . . 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 2222	. . . 4 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` A ) = if ( A = K , K , ( J ` ( A - 1 ) ) ) )
12		f1ocnvfv2 5803	. . . . . . . 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 5484	. . . . . . . . 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 10057	. . . . . . . . . 10 |- ( K e. ( M ... N ) -> K e. ( ZZ>= ` M ) )
19		fzss1 10099	. . . . . . . . . 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 5496	. . . . . . . . . . . 12 |- ( J : ( M ... N ) -1-1-onto-> ( M ... N ) -> `' J : ( M ... N ) -1-1-onto-> ( M ... N ) )
22		f1of 5483	. . . . . . . . . . . 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 5676	. . . . . . . . . 10 |- ( ph -> ( `' J ` K ) e. ( M ... N ) )
25		elfzuz3 10058	. . . . . . . . . 10 |- ( ( `' J ` K ) e. ( M ... N ) -> N e. ( ZZ>= ` ( `' J ` K ) ) )
26		fzss2 10100	. . . . . . . . . 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 3180	. . . . . . . 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 10072	. . . . . . . . 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 5775	. . . . . . 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 1249	. . . . . 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 2267	. . . . 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 10061	. . . . . . . . . 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 10061	. . . . . . . . 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 10061	. . . . . . . . . . 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 9326	. . . . . . . . 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 1179	. . . . . . 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 2191	. . . . . . . . . . 11 |- ( A = K <-> K = A )
54	52, 53	sylnib 677	. . . . . . . . . 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 10063	. . . . . . . . . . . 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 9335	. . . . . . . . . . . 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 1360	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> K < A )
62		zltlem1 9345	. . . . . . . . . 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 9410	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) e. RR )
66	48	zred 9410	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> A e. RR )
67	44	zred 9410	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( `' J ` K ) e. RR )
68	66	lem1d 8925	. . . . . . . . 9 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( A - 1 ) <_ A )
69		elfzle2 10064	. . . . . . . . . 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 8116	. . . . . . . 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 10051	. . . . . . 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 5775	. . . . . 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 1249	. . . . 5 |- ( ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) /\ -. A = K ) -> ( J ` ( A - 1 ) ) e. ( J " ( K ... ( `' J ` K ) ) ) )
77		zdceq 9363	. . . . . 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 3578	. . . 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 2266	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` A ) e. ( J " ( K ... ( `' J ` K ) ) ) )
81	2, 80	eqeltrrd 2267	. 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 10526	. . . . 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 3559	. . . 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 2222	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> ( Q ` B ) = ( J ` B ) )
88		f1of1 5482	. . . . . . 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 5798	. . . . . 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 1249	. . . . 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 674	. . 3 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> -. ( J ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) )
94	87, 93	eqneltrd 2285	. 2 |- ( ( ph /\ ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) ) -> -. ( Q ` B ) e. ( J " ( K ... ( `' J ` K ) ) ) )
95	81, 94	pm2.65da 662	1 |- ( ph -> -. ( A e. ( K ... ( `' J ` K ) ) /\ -. B e. ( K ... ( `' J ` K ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2160   C_ wss 3144   ifcif 3549   class class class wbr 4021   |-> cmpt 4082   `'ccnv 4646   "cima 4650   Fn wfn 5233   -->wf 5234   -1-1->wf1 5235   -1-1-onto->wf1o 5237   ` cfv 5238  (class class class)co 5900   1c1 7847   < clt 8027   <_ cle 8028   - cmin 8163   ZZcz 9288   ZZ>=cuz 9563   ...cfz 10044

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-ltadd 7962

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-inn 8955  df-n0 9212  df-z 9289  df-uz 9564  df-fz 10045

This theorem is referenced by:  iseqf1olemmo  10531