Theorem ledivge1le 9662

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 9661	. . . . . . . . 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 9620	. . . . . . . . . . 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 7914	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> 1 e. RR )
6		rpre 9596	. . . . . . . . . . 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 7981	. . . . . . . . . 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 1228	. . . . . . . . 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 361	. . . . 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 1210	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) /\ A <_ B ) -> ( A / B ) <_ C )
16		simp1 987	. . . . . 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 8025	. . . . . . . . . 10 |- 0 < 1
19		0red 7900	. . . . . . . . . . 11 |- ( C e. RR+ -> 0 e. RR )
20		1red 7914	. . . . . . . . . . 11 |- ( C e. RR+ -> 1 e. RR )
21		ltletr 7988	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ C e. RR ) -> ( ( 0 < 1 /\ 1 <_ C ) -> 0 < C ) )
22	19, 20, 6, 21	syl3anc 1228	. . . . . . . . . 10 |- ( C e. RR+ -> ( ( 0 < 1 /\ 1 <_ C ) -> 0 < C ) )
23	18, 22	mpani 427	. . . . . . . . 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 1010	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( C e. RR /\ 0 < C ) )
27		rpregt0 9603	. . . . . . 7 |- ( B e. RR+ -> ( B e. RR /\ 0 < B ) )
28	27	3ad2ant2 1009	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( B e. RR /\ 0 < B ) )
29	16, 26, 28	3jca 1167	. . . . 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 8788	. . . 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 968   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   1c1 7754   < clt 7933   <_ cle 7934   / cdiv 8568   RR+crp 9589

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-rp 9590

This theorem is referenced by: (None)