Theorem rebtwn2zlemstep 10636

Description: Lemma for rebtwn2z 10638. 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 9630	. . . . . . . 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 9719	. . . . . . . . . 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 8306	. . . . . . . . . 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 9883	. . . . . . . . . . 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 8312	. . . . . . . . 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 8440	. . . . . . . . . 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 6074	. . . . . . . . 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 2267	. . . . . . . 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 4142	. . . . . . 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 4117	. . . . . . . . 9 |- ( j = ( m + 1 ) -> ( j < A <-> ( m + 1 ) < A ) )
16		oveq1 6065	. . . . . . . . . 10 |- ( j = ( m + 1 ) -> ( j + K ) = ( ( m + 1 ) + K ) )
17	16	breq2d 4126	. . . . . . . . 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 2923	. . . . . . 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 4117	. . . . . . . . 9 |- ( j = m -> ( j < A <-> m < A ) )
25		oveq1 6065	. . . . . . . . . 10 |- ( j = m -> ( j + K ) = ( m + K ) )
26	25	breq2d 4126	. . . . . . . . 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 2923	. . . . . . 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 8305	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> 1 e. RR )
31		eluzelre 9882	. . . . . . . . 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 9718	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> m e. RR )
35		1z 9620	. . . . . . . . . . 11 |- 1 e. ZZ
36		eluzp1l 9897	. . . . . . . . . . 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 9313	. . . . . . . . . . 11 |- 2 = ( 1 + 1 )
39	38	fveq2i 5678	. . . . . . . . . 10 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
40	37, 39	eleq2s 2329	. . . . . . . . 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 8713	. . . . . . 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 8319	. . . . . . . 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 8319	. . . . . . . 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 8357	. . . . . . . 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 2662	. . 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 4117	. . . 4 |- ( m = j -> ( m < A <-> j < A ) )
54		oveq1 6065	. . . . 5 |- ( m = j -> ( m + K ) = ( j + K ) )
55	54	breq2d 4126	. . . 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 2781	. 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 2205   E.wrex 2523   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   1c1 8144   + caddc 8146   < clt 8324   2c2 9305   ZZcz 9594   ZZ>=cuz 9871

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872

This theorem is referenced by:  rebtwn2zlemshrink  10637