Theorem ledivge1le 10022

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 10021	. . . . . . . . 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 9980	. . . . . . . . . . 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 8254	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR+ ) /\ C e. RR+ ) -> 1 e. RR )
6		rpre 9956	. . . . . . . . . . 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 8321	. . . . . . . . . 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 1274	. . . . . . . . 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 1247	. . 3 |- ( ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) /\ A <_ B ) -> ( A / B ) <_ C )
16		simp1 1024	. . . . . 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 8365	. . . . . . . . . 10 |- 0 < 1
19		0red 8240	. . . . . . . . . . 11 |- ( C e. RR+ -> 0 e. RR )
20		1red 8254	. . . . . . . . . . 11 |- ( C e. RR+ -> 1 e. RR )
21		ltletr 8328	. . . . . . . . . . 11 |- ( ( 0 e. RR /\ 1 e. RR /\ C e. RR ) -> ( ( 0 < 1 /\ 1 <_ C ) -> 0 < C ) )
22	19, 20, 6, 21	syl3anc 1274	. . . . . . . . . 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 1047	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( C e. RR /\ 0 < C ) )
27		rpregt0 9963	. . . . . . 7 |- ( B e. RR+ -> ( B e. RR /\ 0 < B ) )
28	27	3ad2ant2 1046	. . . . . 6 |- ( ( A e. RR /\ B e. RR+ /\ ( C e. RR+ /\ 1 <_ C ) ) -> ( B e. RR /\ 0 < B ) )
29	16, 26, 28	3jca 1204	. . . . 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 9132	. . . 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 1005   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8091   0cc0 8092   1c1 8093   < clt 8273   <_ cle 8274   / cdiv 8911   RR+crp 9949

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-rp 9950

This theorem is referenced by:  gausslemma2dlem1a  15877