Theorem rebtwn2zlemstep 10575

Description: Lemma for rebtwn2z 10577. 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 9576	. . . . . . . 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 538	. . . . . . . 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 536	. . . . . . . . . . 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 9664	. . . . . . . . . 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 8255	. . . . . . . . . 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 9828	. . . . . . . . . . 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 8261	. . . . . . . . 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 8389	. . . . . . . . . 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 6044	. . . . . . . . 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 2264	. . . . . . . 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 4121	. . . . . . 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 4096	. . . . . . . . 9 |- ( j = ( m + 1 ) -> ( j < A <-> ( m + 1 ) < A ) )
16		oveq1 6035	. . . . . . . . . 10 |- ( j = ( m + 1 ) -> ( j + K ) = ( ( m + 1 ) + K ) )
17	16	breq2d 4105	. . . . . . . . 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 2911	. . . . . . 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 1272	. . . . . 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 536	. . . . . . 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 537	. . . . . . 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 4096	. . . . . . . . 9 |- ( j = m -> ( j < A <-> m < A ) )
25		oveq1 6035	. . . . . . . . . 10 |- ( j = m -> ( j + K ) = ( m + K ) )
26	25	breq2d 4105	. . . . . . . . 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 2911	. . . . . . 7 |- ( ( m e. ZZ /\ ( m < A /\ A < ( m + K ) ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
29	21, 22, 23, 28	syl12anc 1272	. . . . . 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 8254	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> 1 e. RR )
31		eluzelre 9827	. . . . . . . . 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 529	. . . . . . . . 9 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> m e. ZZ )
34	33	zred 9663	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> m e. RR )
35		1z 9566	. . . . . . . . . . 11 |- 1 e. ZZ
36		eluzp1l 9842	. . . . . . . . . . 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 9261	. . . . . . . . . . 11 |- 2 = ( 1 + 1 )
39	38	fveq2i 5651	. . . . . . . . . 10 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
40	37, 39	eleq2s 2326	. . . . . . . . 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 8661	. . . . . . 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 8268	. . . . . . . 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 8268	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> ( m + K ) e. RR )
45		simpllr 536	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> A e. RR )
46		axltwlin 8306	. . . . . . . 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 1274	. . . . . . 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 806	. . . . 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 2651	. . 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 1227	. 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 4096	. . . 4 |- ( m = j -> ( m < A <-> j < A ) )
54		oveq1 6035	. . . . 5 |- ( m = j -> ( m + K ) = ( j + K ) )
55	54	breq2d 4105	. . . 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 2769	. 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 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   E.wrex 2512   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   RRcr 8091   1c1 8093   + caddc 8095   < clt 8273   2c2 9253   ZZcz 9540   ZZ>=cuz 9816

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-inn 9203  df-2 9261  df-n0 9462  df-z 9541  df-uz 9817

This theorem is referenced by:  rebtwn2zlemshrink  10576