Theorem ledivge1le 9513

Description: If a number is less than or equal to another number, the number divided by a positive number greater than or equal to one is less than or equal to the other number. (Contributed by AV, 29-Jun-2021.)

Assertion

Ref	Expression
ledivge1le	|- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( A <_ B -> ( A / C ) <_ B ) )

Proof of Theorem ledivge1le

Step	Hyp	Ref	Expression
1		divle1le 9512	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) <_ 1 <-> A <_ B ) )
2	1	adantr 274	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( ( A / B ) <_ 1 <-> A <_ B ) )
3		rerpdivcl 9472	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
4	3	adantr 274	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( A / B ) e. RR )
5		1red 7781	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> 1 e. RR )
6		rpre 9448	. . . . . . . . . . 11 |- ( C e. RR+ -> C e. RR )
7	6	adantl 275	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> C e. RR )
8		letr 7847	. . . . . . . . . 10 |- ( ( ( A / B ) e. RR /\ 1 e. RR /\ C e. RR ) -> ( ( ( A / B ) <_ 1 /\ 1 <_ C ) -> ( A / B ) <_ C ) )
9	4, 5, 7, 8	syl3anc 1216	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( ( ( A / B ) <_ 1 /\ 1 <_ C ) -> ( A / B ) <_ C ) )
10	9	expd 256	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( ( A / B ) <_ 1 -> ( 1 <_ C -> ( A / B ) <_ C ) ) )
11	2, 10	sylbird 169	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( A <_ B -> ( 1 <_ C -> ( A / B ) <_ C ) ) )
12	11	com23 78	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( 1 <_ C -> ( A <_ B -> ( A / B ) <_ C ) ) )
13	12	expimpd 360	. . . . 5 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( C e. RR+ /\ 1 <_ C ) -> ( A <_ B -> ( A / B ) <_ C ) ) )
14	13	ex 114	. . . 4 |- ( A e. RR -> ( B e. RR+ -> ( ( C e. RR+ /\ 1 <_ C ) -> ( A <_ B -> ( A / B ) <_ C ) ) ) )
15	14	3imp1 1198	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) /\ A <_ B ) -> ( A / B ) <_ C )
16		simp1 981	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> A e. RR )
17	6	adantr 274	. . . . . . . 8 |- ( ( C e. RR+ /\ 1 <_ C ) -> C e. RR )
18		0lt1 7889	. . . . . . . . . 10 |- 0 < 1
19		0red 7767	. . . . . . . . . . 11 |- ( C e. RR+ -> 0 e. RR )
20		1red 7781	. . . . . . . . . . 11 |- ( C e. RR+ -> 1 e. RR )
21		ltletr 7853	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ C e. RR ) -> ( ( 0 < 1 /\ 1 <_ C ) -> 0 < C ) )
22	19, 20, 6, 21	syl3anc 1216	. . . . . . . . . 10 |- ( C e. RR+ -> ( ( 0 < 1 /\ 1 <_ C ) -> 0 < C ) )
23	18, 22	mpani 426	. . . . . . . . 9 |- ( C e. RR+ -> ( 1 <_ C -> 0 < C ) )
24	23	imp 123	. . . . . . . 8 |- ( ( C e. RR+ /\ 1 <_ C ) -> 0 < C )
25	17, 24	jca 304	. . . . . . 7 |- ( ( C e. RR+ /\ 1 <_ C ) -> ( C e. RR /\ 0 < C ) )
26	25	3ad2ant3 1004	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( C e. RR /\ 0 < C ) )
27		rpregt0 9455	. . . . . . 7 |- ( B e. RR+ -> ( B e. RR /\ 0 < B ) )
28	27	3ad2ant2 1003	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( B e. RR /\ 0 < B ) )
29	16, 26, 28	3jca 1161	. . . . 5 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( A e. RR /\ ( C e. RR /\ 0 < C ) /\ ( B e. RR /\ 0 < B ) ) )
30	29	adantr 274	. . . 4 |- ( ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) /\ A <_ B ) -> ( A e. RR /\ ( C e. RR /\ 0 < C ) /\ ( B e. RR /\ 0 < B ) ) )
31		lediv23 8651	. . . 4 |- ( ( A e. RR /\ ( C e. RR /\ 0 < C ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / C ) <_ B <-> ( A / B ) <_ C ) )
32	30, 31	syl 14	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) /\ A <_ B ) -> ( ( A / C ) <_ B <-> ( A / B ) <_ C ) )
33	15, 32	mpbird 166	. 2 |- ( ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) /\ A <_ B ) -> ( A / C ) <_ B )
34	33	ex 114	1 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( A <_ B -> ( A / C ) <_ B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7619   0cc0 7620   1c1 7621   < clt 7800   <_ cle 7801   / cdiv 8432   RR+crp 9441

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-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  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-rmo 2424  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-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  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-reap 8337  df-ap 8344  df-div 8433  df-rp 9442

This theorem is referenced by: (None)