Theorem ledivge1le 9724

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 9723	. . . . . . . . 9 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) <_ 1 <-> A <_ B ) )
2	1	adantr 276	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( ( A / B ) <_ 1 <-> A <_ B ) )
3		rerpdivcl 9682	. . . . . . . . . . 11 |- ( ( A e. RR /\ B e. RR+ ) -> ( A / B ) e. RR )
4	3	adantr 276	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( A / B ) e. RR )
5		1red 7971	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> 1 e. RR )
6		rpre 9658	. . . . . . . . . . 11 |- ( C e. RR+ -> C e. RR )
7	6	adantl 277	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> C e. RR )
8		letr 8038	. . . . . . . . . 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 1238	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( ( ( A / B ) <_ 1 /\ 1 <_ C ) -> ( A / B ) <_ C ) )
10	9	expd 258	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> ( ( A / B ) <_ 1 -> ( 1 <_ C -> ( A / B ) <_ C ) ) )
11	2, 10	sylbird 170	. . . . . . 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 363	. . . . 5 |- ( ( A e. RR /\ B e. RR+ ) -> ( ( C e. RR+ /\ 1 <_ C ) -> ( A <_ B -> ( A / B ) <_ C ) ) )
14	13	ex 115	. . . 4 |- ( A e. RR -> ( B e. RR+ -> ( ( C e. RR+ /\ 1 <_ C ) -> ( A <_ B -> ( A / B ) <_ C ) ) ) )
15	14	3imp1 1220	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) /\ A <_ B ) -> ( A / B ) <_ C )
16		simp1 997	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> A e. RR )
17	6	adantr 276	. . . . . . . 8 |- ( ( C e. RR+ /\ 1 <_ C ) -> C e. RR )
18		0lt1 8082	. . . . . . . . . 10 |- 0 < 1
19		0red 7957	. . . . . . . . . . 11 |- ( C e. RR+ -> 0 e. RR )
20		1red 7971	. . . . . . . . . . 11 |- ( C e. RR+ -> 1 e. RR )
21		ltletr 8045	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ C e. RR ) -> ( ( 0 < 1 /\ 1 <_ C ) -> 0 < C ) )
22	19, 20, 6, 21	syl3anc 1238	. . . . . . . . . 10 |- ( C e. RR+ -> ( ( 0 < 1 /\ 1 <_ C ) -> 0 < C ) )
23	18, 22	mpani 430	. . . . . . . . 9 |- ( C e. RR+ -> ( 1 <_ C -> 0 < C ) )
24	23	imp 124	. . . . . . . 8 |- ( ( C e. RR+ /\ 1 <_ C ) -> 0 < C )
25	17, 24	jca 306	. . . . . . 7 |- ( ( C e. RR+ /\ 1 <_ C ) -> ( C e. RR /\ 0 < C ) )
26	25	3ad2ant3 1020	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( C e. RR /\ 0 < C ) )
27		rpregt0 9665	. . . . . . 7 |- ( B e. RR+ -> ( B e. RR /\ 0 < B ) )
28	27	3ad2ant2 1019	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( B e. RR /\ 0 < B ) )
29	16, 26, 28	3jca 1177	. . . . 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 276	. . . 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 8848	. . . 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 167	. 2 |- ( ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) /\ A <_ B ) -> ( A / C ) <_ B )
34	33	ex 115	1 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( A <_ B -> ( A / C ) <_ B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 978   e. wcel 2148   class class class wbr 4003  (class class class)co 5874   RRcr 7809   0cc0 7810   1c1 7811   < clt 7990   <_ cle 7991   / cdiv 8627   RR+crp 9651

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-po 4296  df-iso 4297  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-rp 9652

This theorem is referenced by: (None)