Theorem rebtwn2zlemstep 10502

Description: Lemma for rebtwn2z 10504. Induction step. (Contributed by Jim Kingdon, 13-Oct-2021.)

Assertion

Ref	Expression
rebtwn2zlemstep	|- ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR /\ E. m e. ZZ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> E. m e. ZZ ( m < A /\ A < ( m + K ) ) )

Distinct variable groups:   A, m   m, K

Proof of Theorem rebtwn2zlemstep

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2z 9505	. . . . . . . 8 |- ( m e. ZZ -> ( m + 1 ) e. ZZ )
2	1	ad3antlr 493	. . . . . . 7 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> ( m + 1 ) e. ZZ )
3		simpr 110	. . . . . . 7 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> ( m + 1 ) < A )
4		simplrr 536	. . . . . . . 8 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> A < ( m + ( K + 1 ) ) )
5		simpllr 534	. . . . . . . . . . 11 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> m e. ZZ )
6	5	zcnd 9593	. . . . . . . . . 10 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> m e. CC )
7		1cnd 8185	. . . . . . . . . 10 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> 1 e. CC )
8		eluzelcn 9757	. . . . . . . . . . 11 |- ( K e. ( ZZ>= ` 2 ) -> K e. CC )
9	8	ad4antr 494	. . . . . . . . . 10 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> K e. CC )
10	6, 7, 9	addassd 8192	. . . . . . . . 9 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> ( ( m + 1 ) + K ) = ( m + ( 1 + K ) ) )
11	7, 9	addcomd 8320	. . . . . . . . . 10 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> ( 1 + K ) = ( K + 1 ) )
12	11	oveq2d 6029	. . . . . . . . 9 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> ( m + ( 1 + K ) ) = ( m + ( K + 1 ) ) )
13	10, 12	eqtrd 2262	. . . . . . . 8 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> ( ( m + 1 ) + K ) = ( m + ( K + 1 ) ) )
14	4, 13	breqtrrd 4114	. . . . . . 7 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> A < ( ( m + 1 ) + K ) )
15		breq1 4089	. . . . . . . . 9 |- ( j = ( m + 1 ) -> ( j < A <-> ( m + 1 ) < A ) )
16		oveq1 6020	. . . . . . . . . 10 |- ( j = ( m + 1 ) -> ( j + K ) = ( ( m + 1 ) + K ) )
17	16	breq2d 4098	. . . . . . . . 9 |- ( j = ( m + 1 ) -> ( A < ( j + K ) <-> A < ( ( m + 1 ) + K ) ) )
18	15, 17	anbi12d 473	. . . . . . . 8 |- ( j = ( m + 1 ) -> ( ( j < A /\ A < ( j + K ) ) <-> ( ( m + 1 ) < A /\ A < ( ( m + 1 ) + K ) ) ) )
19	18	rspcev 2908	. . . . . . 7 |- ( ( ( m + 1 ) e. ZZ /\ ( ( m + 1 ) < A /\ A < ( ( m + 1 ) + K ) ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
20	2, 3, 14, 19	syl12anc 1269	. . . . . 6 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ ( m + 1 ) < A ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
21		simpllr 534	. . . . . . 7 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ A < ( m + K ) ) -> m e. ZZ )
22		simplrl 535	. . . . . . 7 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ A < ( m + K ) ) -> m < A )
23		simpr 110	. . . . . . 7 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ A < ( m + K ) ) -> A < ( m + K ) )
24		breq1 4089	. . . . . . . . 9 |- ( j = m -> ( j < A <-> m < A ) )
25		oveq1 6020	. . . . . . . . . 10 |- ( j = m -> ( j + K ) = ( m + K ) )
26	25	breq2d 4098	. . . . . . . . 9 |- ( j = m -> ( A < ( j + K ) <-> A < ( m + K ) ) )
27	24, 26	anbi12d 473	. . . . . . . 8 |- ( j = m -> ( ( j < A /\ A < ( j + K ) ) <-> ( m < A /\ A < ( m + K ) ) ) )
28	27	rspcev 2908	. . . . . . 7 |- ( ( m e. ZZ /\ ( m < A /\ A < ( m + K ) ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
29	21, 22, 23, 28	syl12anc 1269	. . . . . 6 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ A < ( m + K ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
30		1red 8184	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> 1 e. RR )
31		eluzelre 9756	. . . . . . . . 9 |- ( K e. ( ZZ>= ` 2 ) -> K e. RR )
32	31	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> K e. RR )
33		simplr 528	. . . . . . . . 9 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> m e. ZZ )
34	33	zred 9592	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> m e. RR )
35		1z 9495	. . . . . . . . . . 11 |- 1 e. ZZ
36		eluzp1l 9771	. . . . . . . . . . 11 |- ( ( 1 e. ZZ /\ K e. ( ZZ>= ` ( 1 + 1 ) ) ) -> 1 < K )
37	35, 36	mpan 424	. . . . . . . . . 10 |- ( K e. ( ZZ>= ` ( 1 + 1 ) ) -> 1 < K )
38		df-2 9192	. . . . . . . . . . 11 |- 2 = ( 1 + 1 )
39	38	fveq2i 5638	. . . . . . . . . 10 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
40	37, 39	eleq2s 2324	. . . . . . . . 9 |- ( K e. ( ZZ>= ` 2 ) -> 1 < K )
41	40	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> 1 < K )
42	30, 32, 34, 41	ltadd2dd 8592	. . . . . . 7 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> ( m + 1 ) < ( m + K ) )
43	34, 30	readdcld 8199	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> ( m + 1 ) e. RR )
44	34, 32	readdcld 8199	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> ( m + K ) e. RR )
45		simpllr 534	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> A e. RR )
46		axltwlin 8237	. . . . . . . 8 |- ( ( ( m + 1 ) e. RR /\ ( m + K ) e. RR /\ A e. RR ) -> ( ( m + 1 ) < ( m + K ) -> ( ( m + 1 ) < A \/ A < ( m + K ) ) ) )
47	43, 44, 45, 46	syl3anc 1271	. . . . . . 7 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> ( ( m + 1 ) < ( m + K ) -> ( ( m + 1 ) < A \/ A < ( m + K ) ) ) )
48	42, 47	mpd 13	. . . . . 6 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> ( ( m + 1 ) < A \/ A < ( m + K ) ) )
49	20, 29, 48	mpjaodan 803	. . . . 5 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
50	49	ex 115	. . . 4 |- ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) -> ( ( m < A /\ A < ( m + ( K + 1 ) ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) ) )
51	50	rexlimdva 2648	. . 3 |- ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) -> ( E. m e. ZZ ( m < A /\ A < ( m + ( K + 1 ) ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) ) )
52	51	3impia 1224	. 2 |- ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR /\ E. m e. ZZ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
53		breq1 4089	. . . 4 |- ( m = j -> ( m < A <-> j < A ) )
54		oveq1 6020	. . . . 5 |- ( m = j -> ( m + K ) = ( j + K ) )
55	54	breq2d 4098	. . . 4 |- ( m = j -> ( A < ( m + K ) <-> A < ( j + K ) ) )
56	53, 55	anbi12d 473	. . 3 |- ( m = j -> ( ( m < A /\ A < ( m + K ) ) <-> ( j < A /\ A < ( j + K ) ) ) )
57	56	cbvrexv 2766	. 2 |- ( E. m e. ZZ ( m < A /\ A < ( m + K ) ) <-> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
58	52, 57	sylibr 134	1 |- ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR /\ E. m e. ZZ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> E. m e. ZZ ( m < A /\ A < ( m + K ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8020   RRcr 8021   1c1 8023   + caddc 8025   < clt 8204   2c2 9184   ZZcz 9469   ZZ>=cuz 9745

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-2 9192  df-n0 9393  df-z 9470  df-uz 9746

This theorem is referenced by:  rebtwn2zlemshrink  10503