Theorem rebtwn2zlemstep 10030

Description: Lemma for rebtwn2z 10032. 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 9090	. . . . . . . 8 |- ( m e. ZZ -> ( m + 1 ) e. ZZ )
2	1	ad3antlr 484	. . . . . . 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 109	. . . . . . 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 525	. . . . . . . 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 523	. . . . . . . . . . 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 9174	. . . . . . . . . 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 7782	. . . . . . . . . 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 9337	. . . . . . . . . . 11 |- ( K e. ( ZZ>= ` 2 ) -> K e. CC )
9	8	ad4antr 485	. . . . . . . . . 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 7788	. . . . . . . . 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 7913	. . . . . . . . . 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 5790	. . . . . . . . 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 2172	. . . . . . . 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 3956	. . . . . . 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 3932	. . . . . . . . 9 |- ( j = ( m + 1 ) -> ( j < A <-> ( m + 1 ) < A ) )
16		oveq1 5781	. . . . . . . . . 10 |- ( j = ( m + 1 ) -> ( j + K ) = ( ( m + 1 ) + K ) )
17	16	breq2d 3941	. . . . . . . . 9 |- ( j = ( m + 1 ) -> ( A < ( j + K ) <-> A < ( ( m + 1 ) + K ) ) )
18	15, 17	anbi12d 464	. . . . . . . 8 |- ( j = ( m + 1 ) -> ( ( j < A /\ A < ( j + K ) ) <-> ( ( m + 1 ) < A /\ A < ( ( m + 1 ) + K ) ) ) )
19	18	rspcev 2789	. . . . . . 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 1214	. . . . . 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 523	. . . . . . 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 524	. . . . . . 7 |- ( ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) /\ A < ( m + K ) ) -> m < A )
23		simpr 109	. . . . . . 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 3932	. . . . . . . . 9 |- ( j = m -> ( j < A <-> m < A ) )
25		oveq1 5781	. . . . . . . . . 10 |- ( j = m -> ( j + K ) = ( m + K ) )
26	25	breq2d 3941	. . . . . . . . 9 |- ( j = m -> ( A < ( j + K ) <-> A < ( m + K ) ) )
27	24, 26	anbi12d 464	. . . . . . . 8 |- ( j = m -> ( ( j < A /\ A < ( j + K ) ) <-> ( m < A /\ A < ( m + K ) ) ) )
28	27	rspcev 2789	. . . . . . 7 |- ( ( m e. ZZ /\ ( m < A /\ A < ( m + K ) ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
29	21, 22, 23, 28	syl12anc 1214	. . . . . 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 7781	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> 1 e. RR )
31		eluzelre 9336	. . . . . . . . 9 |- ( K e. ( ZZ>= ` 2 ) -> K e. RR )
32	31	ad3antrrr 483	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> K e. RR )
33		simplr 519	. . . . . . . . 9 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> m e. ZZ )
34	33	zred 9173	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> m e. RR )
35		1z 9080	. . . . . . . . . . 11 |- 1 e. ZZ
36		eluzp1l 9350	. . . . . . . . . . 11 |- ( ( 1 e. ZZ /\ K e. ( ZZ>= ` ( 1 + 1 ) ) ) -> 1 < K )
37	35, 36	mpan 420	. . . . . . . . . 10 |- ( K e. ( ZZ>= ` ( 1 + 1 ) ) -> 1 < K )
38		df-2 8779	. . . . . . . . . . 11 |- 2 = ( 1 + 1 )
39	38	fveq2i 5424	. . . . . . . . . 10 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
40	37, 39	eleq2s 2234	. . . . . . . . 9 |- ( K e. ( ZZ>= ` 2 ) -> 1 < K )
41	40	ad3antrrr 483	. . . . . . . 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 8184	. . . . . . 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 7795	. . . . . . . 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 7795	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> ( m + K ) e. RR )
45		simpllr 523	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> A e. RR )
46		axltwlin 7832	. . . . . . . 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 1216	. . . . . . 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 787	. . . . 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 114	. . . 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 2549	. . 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 1178	. 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 3932	. . . 4 |- ( m = j -> ( m < A <-> j < A ) )
54		oveq1 5781	. . . . 5 |- ( m = j -> ( m + K ) = ( j + K ) )
55	54	breq2d 3941	. . . 4 |- ( m = j -> ( A < ( m + K ) <-> A < ( j + K ) ) )
56	53, 55	anbi12d 464	. . 3 |- ( m = j -> ( ( m < A /\ A < ( m + K ) ) <-> ( j < A /\ A < ( j + K ) ) ) )
57	56	cbvrexv 2655	. 2 |- ( E. m e. ZZ ( m < A /\ A < ( m + K ) ) <-> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
58	52, 57	sylibr 133	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 103   \/ wo 697   /\ w3a 962   = wceq 1331   e. wcel 1480   E.wrex 2417   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7618   RRcr 7619   1c1 7621   + caddc 7623   < clt 7800   2c2 8771   ZZcz 9054   ZZ>=cuz 9326

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-2 8779  df-n0 8978  df-z 9055  df-uz 9327

This theorem is referenced by:  rebtwn2zlemshrink  10031