Theorem rebtwn2zlemstep 10665

Description: Lemma for rebtwn2z 10667. 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 9659	. . . . . . . 8 |- ( m e. ZZ -> ( m + 1 ) e. ZZ )
2	1	ad3antlr 497	. . . . . . 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 542	. . . . . . . 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 540	. . . . . . . . . . 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 9748	. . . . . . . . . 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 8332	. . . . . . . . . 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 9912	. . . . . . . . . . 11 |- ( K e. ( ZZ>= ` 2 ) -> K e. CC )
9	8	ad4antr 498	. . . . . . . . . 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 8338	. . . . . . . . 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 8467	. . . . . . . . . 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 6091	. . . . . . . . 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 2271	. . . . . . . 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 4153	. . . . . . 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 4128	. . . . . . . . 9 |- ( j = ( m + 1 ) -> ( j < A <-> ( m + 1 ) < A ) )
16		oveq1 6082	. . . . . . . . . 10 |- ( j = ( m + 1 ) -> ( j + K ) = ( ( m + 1 ) + K ) )
17	16	breq2d 4137	. . . . . . . . 9 |- ( j = ( m + 1 ) -> ( A < ( j + K ) <-> A < ( ( m + 1 ) + K ) ) )
18	15, 17	anbi12d 477	. . . . . . . 8 |- ( j = ( m + 1 ) -> ( ( j < A /\ A < ( j + K ) ) <-> ( ( m + 1 ) < A /\ A < ( ( m + 1 ) + K ) ) ) )
19	18	rspcev 2929	. . . . . . 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 1276	. . . . . 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 540	. . . . . . 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 541	. . . . . . 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 4128	. . . . . . . . 9 |- ( j = m -> ( j < A <-> m < A ) )
25		oveq1 6082	. . . . . . . . . 10 |- ( j = m -> ( j + K ) = ( m + K ) )
26	25	breq2d 4137	. . . . . . . . 9 |- ( j = m -> ( A < ( j + K ) <-> A < ( m + K ) ) )
27	24, 26	anbi12d 477	. . . . . . . 8 |- ( j = m -> ( ( j < A /\ A < ( j + K ) ) <-> ( m < A /\ A < ( m + K ) ) ) )
28	27	rspcev 2929	. . . . . . 7 |- ( ( m e. ZZ /\ ( m < A /\ A < ( m + K ) ) ) -> E. j e. ZZ ( j < A /\ A < ( j + K ) ) )
29	21, 22, 23, 28	syl12anc 1276	. . . . . 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 8331	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> 1 e. RR )
31		eluzelre 9911	. . . . . . . . 9 |- ( K e. ( ZZ>= ` 2 ) -> K e. RR )
32	31	ad3antrrr 496	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> K e. RR )
33		simplr 533	. . . . . . . . 9 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> m e. ZZ )
34	33	zred 9747	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> m e. RR )
35		1z 9649	. . . . . . . . . . 11 |- 1 e. ZZ
36		eluzp1l 9926	. . . . . . . . . . 11 |- ( ( 1 e. ZZ /\ K e. ( ZZ>= ` ( 1 + 1 ) ) ) -> 1 < K )
37	35, 36	mpan 428	. . . . . . . . . 10 |- ( K e. ( ZZ>= ` ( 1 + 1 ) ) -> 1 < K )
38		df-2 9342	. . . . . . . . . . 11 |- 2 = ( 1 + 1 )
39	38	fveq2i 5693	. . . . . . . . . 10 |- ( ZZ>= ` 2 ) = ( ZZ>= ` ( 1 + 1 ) )
40	37, 39	eleq2s 2333	. . . . . . . . 9 |- ( K e. ( ZZ>= ` 2 ) -> 1 < K )
41	40	ad3antrrr 496	. . . . . . . 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 8740	. . . . . . 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 8345	. . . . . . . 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 8345	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> ( m + K ) e. RR )
45		simpllr 540	. . . . . . . 8 |- ( ( ( ( K e. ( ZZ>= ` 2 ) /\ A e. RR ) /\ m e. ZZ ) /\ ( m < A /\ A < ( m + ( K + 1 ) ) ) ) -> A e. RR )
46		axltwlin 8383	. . . . . . . 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 1278	. . . . . . 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 810	. . . . 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 2668	. . 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 1231	. 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 4128	. . . 4 |- ( m = j -> ( m < A <-> j < A ) )
54		oveq1 6082	. . . . 5 |- ( m = j -> ( m + K ) = ( j + K ) )
55	54	breq2d 4137	. . . 4 |- ( m = j -> ( A < ( m + K ) <-> A < ( j + K ) ) )
56	53, 55	anbi12d 477	. . 3 |- ( m = j -> ( ( m < A /\ A < ( m + K ) ) <-> ( j < A /\ A < ( j + K ) ) ) )
57	56	cbvrexv 2787	. 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 720   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   1c1 8170   + caddc 8172   < clt 8350   2c2 9334   ZZcz 9623   ZZ>=cuz 9900

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901

This theorem is referenced by:  rebtwn2zlemshrink  10666