Theorem ledivge1le 9792

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 9791	. . . . . . . . 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 9750	. . . . . . . . . . 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 8034	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> 1 e. RR )
6		rpre 9726	. . . . . . . . . . 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 8102	. . . . . . . . . 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 1249	. . . . . . . . 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 1222	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) /\ A <_ B ) -> ( A / B ) <_ C )
16		simp1 999	. . . . . 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 8146	. . . . . . . . . 10 |- 0 < 1
19		0red 8020	. . . . . . . . . . 11 |- ( C e. RR+ -> 0 e. RR )
20		1red 8034	. . . . . . . . . . 11 |- ( C e. RR+ -> 1 e. RR )
21		ltletr 8109	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ C e. RR ) -> ( ( 0 < 1 /\ 1 <_ C ) -> 0 < C ) )
22	19, 20, 6, 21	syl3anc 1249	. . . . . . . . . 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 1022	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( C e. RR /\ 0 < C ) )
27		rpregt0 9733	. . . . . . 7 |- ( B e. RR+ -> ( B e. RR /\ 0 < B ) )
28	27	3ad2ant2 1021	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( B e. RR /\ 0 < B ) )
29	16, 26, 28	3jca 1179	. . . . 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 8912	. . . 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 980   e. wcel 2164   class class class wbr 4029  (class class class)co 5918   RRcr 7871   0cc0 7872   1c1 7873   < clt 8054   <_ cle 8055   / cdiv 8691   RR+crp 9719

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-rp 9720

This theorem is referenced by:  gausslemma2dlem1a  15174