Theorem ltdiv1t 5851

Description: Division of both sides of 'less than' by a positive number.

Assertion

Ref	Expression
ltdiv1t	|- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> (A < B <-> (A / C) < (B / C)))

Proof of Theorem ltdiv1t

Step	Hyp	Ref	Expression
1		breq1 2627	. . . . . . 7 |- (A = if(A e. RR, A, 0) -> (A < B <-> if(A e. RR, A, 0) < B))
2		opreq1 3974	. . . . . . . 8 |- (A = if(A e. RR, A, 0) -> (A / C) = (if(A e. RR, A, 0) / C))
3	2	breq1d 2634	. . . . . . 7 |- (A = if(A e. RR, A, 0) -> ((A / C) < (B / C) <-> (if(A e. RR, A, 0) / C) < (B / C)))
4	1, 3	bibi12d 631	. . . . . 6 |- (A = if(A e. RR, A, 0) -> ((A < B <-> (A / C) < (B / C)) <-> (if(A e. RR, A, 0) < B <-> (if(A e. RR, A, 0) / C) < (B / C))))
5	4	imbi2d 614	. . . . 5 |- (A = if(A e. RR, A, 0) -> ((0 < C -> (A < B <-> (A / C) < (B / C))) <-> (0 < C -> (if(A e. RR, A, 0) < B <-> (if(A e. RR, A, 0) / C) < (B / C)))))
6		breq2 2628	. . . . . . 7 |- (B = if(B e. RR, B, 0) -> (if(A e. RR, A, 0) < B <-> if(A e. RR, A, 0) < if(B e. RR, B, 0)))
7		opreq1 3974	. . . . . . . 8 |- (B = if(B e. RR, B, 0) -> (B / C) = (if(B e. RR, B, 0) / C))
8	7	breq2d 2635	. . . . . . 7 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) / C) < (B / C) <-> (if(A e. RR, A, 0) / C) < (if(B e. RR, B, 0) / C)))
9	6, 8	bibi12d 631	. . . . . 6 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) < B <-> (if(A e. RR, A, 0) / C) < (B / C)) <-> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) / C) < (if(B e. RR, B, 0) / C))))
10	9	imbi2d 614	. . . . 5 |- (B = if(B e. RR, B, 0) -> ((0 < C -> (if(A e. RR, A, 0) < B <-> (if(A e. RR, A, 0) / C) < (B / C))) <-> (0 < C -> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) / C) < (if(B e. RR, B, 0) / C)))))
11		breq2 2628	. . . . . 6 |- (C = if(C e. RR, C, 0) -> (0 < C <-> 0 < if(C e. RR, C, 0)))
12		opreq2 3975	. . . . . . . 8 |- (C = if(C e. RR, C, 0) -> (if(A e. RR, A, 0) / C) = (if(A e. RR, A, 0) / if(C e. RR, C, 0)))
13		opreq2 3975	. . . . . . . 8 |- (C = if(C e. RR, C, 0) -> (if(B e. RR, B, 0) / C) = (if(B e. RR, B, 0) / if(C e. RR, C, 0)))
14	12, 13	breq12d 2636	. . . . . . 7 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) / C) < (if(B e. RR, B, 0) / C) <-> (if(A e. RR, A, 0) / if(C e. RR, C, 0)) < (if(B e. RR, B, 0) / if(C e. RR, C, 0))))
15	14	bibi2d 620	. . . . . 6 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) / C) < (if(B e. RR, B, 0) / C)) <-> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) / if(C e. RR, C, 0)) < (if(B e. RR, B, 0) / if(C e. RR, C, 0)))))
16	11, 15	imbi12d 628	. . . . 5 |- (C = if(C e. RR, C, 0) -> ((0 < C -> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) / C) < (if(B e. RR, B, 0) / C))) <-> (0 < if(C e. RR, C, 0) -> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) / if(C e. RR, C, 0)) < (if(B e. RR, B, 0) / if(C e. RR, C, 0))))))
17		0re 5452	. . . . . . 7 |- 0 e. RR
18	17	elimel 2398	. . . . . 6 |- if(A e. RR, A, 0) e. RR
19	17	elimel 2398	. . . . . 6 |- if(B e. RR, B, 0) e. RR
20	17	elimel 2398	. . . . . 6 |- if(C e. RR, C, 0) e. RR
21	18, 19, 20	ltdiv1 5826	. . . . 5 |- (0 < if(C e. RR, C, 0) -> (if(A e. RR, A, 0) < if(B e. RR, B, 0) <-> (if(A e. RR, A, 0) / if(C e. RR, C, 0)) < (if(B e. RR, B, 0) / if(C e. RR, C, 0))))
22	5, 10, 16, 21	dedth3h 2392	. . . 4 |- ((A e. RR /\ B e. RR /\ C e. RR) -> (0 < C -> (A < B <-> (A / C) < (B / C))))
23	22	3exp 834	. . 3 |- (A e. RR -> (B e. RR -> (C e. RR -> (0 < C -> (A < B <-> (A / C) < (B / C))))))
24	23	imp4a 364	. 2 |- (A e. RR -> (B e. RR -> ((C e. RR /\ 0 < C) -> (A < B <-> (A / C) < (B / C)))))
25	24	3imp 829	1 |- ((A e. RR /\ B e. RR /\ (C e. RR /\ 0 < C)) -> (A < B <-> (A / C) < (B / C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  ifcif 2365   class class class wbr 2624  (class class class)co 3969  RRcr 5245  0cc0 5246   / cdiv 5306   < clt 5498

This theorem is referenced by:  lediv1t 5853  ltdivmult 5869  quoremz 6253  fldivt 6256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-div 5715

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