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Theorem lt2add 6827
Description: Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20.
Assertion
Ref Expression
lt2add |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A < C /\ B < D) -> (A + B) < (C + D)))
Proof of Theorem lt2add
StepHypRef Expression
1 breq1 3341 . . . 4 |- (A = if(A e. RR, A, 0) -> (A < C <-> if(A e. RR, A, 0) < C))
21anbi1d 679 . . 3 |- (A = if(A e. RR, A, 0) -> ((A < C /\ B < D) <-> (if(A e. RR, A, 0) < C /\ B < D)))
3 opreq1 4889 . . . 4 |- (A = if(A e. RR, A, 0) -> (A + B) = (if(A e. RR, A, 0) + B))
43breq1d 3348 . . 3 |- (A = if(A e. RR, A, 0) -> ((A + B) < (C + D) <-> (if(A e. RR, A, 0) + B) < (C + D)))
52, 4imbi12d 688 . 2 |- (A = if(A e. RR, A, 0) -> (((A < C /\ B < D) -> (A + B) < (C + D)) <-> ((if(A e. RR, A, 0) < C /\ B < D) -> (if(A e. RR, A, 0) + B) < (C + D))))
6 breq1 3341 . . . 4 |- (B = if(B e. RR, B, 0) -> (B < D <-> if(B e. RR, B, 0) < D))
76anbi2d 678 . . 3 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) < C /\ B < D) <-> (if(A e. RR, A, 0) < C /\ if(B e. RR, B, 0) < D)))
8 opreq2 4890 . . . 4 |- (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)))
98breq1d 3348 . . 3 |- (B = if(B e. RR, B, 0) -> ((if(A e. RR, A, 0) + B) < (C + D) <-> (if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (C + D)))
107, 9imbi12d 688 . 2 |- (B = if(B e. RR, B, 0) -> (((if(A e. RR, A, 0) < C /\ B < D) -> (if(A e. RR, A, 0) + B) < (C + D)) <-> ((if(A e. RR, A, 0) < C /\ if(B e. RR, B, 0) < D) -> (if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (C + D))))
11 breq2 3342 . . . 4 |- (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)))
1211anbi1d 679 . . 3 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) < C /\ if(B e. RR, B, 0) < D) <-> (if(A e. RR, A, 0) < if(C e. RR, C, 0) /\ if(B e. RR, B, 0) < D)))
13 opreq1 4889 . . . 4 |- (C = if(C e. RR, C, 0) -> (C + D) = (if(C e. RR, C, 0) + D))
1413breq2d 3350 . . 3 |- (C = if(C e. RR, C, 0) -> ((if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (C + D) <-> (if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (if(C e. RR, C, 0) + D)))
1512, 14imbi12d 688 . 2 |- (C = if(C e. RR, C, 0) -> (((if(A e. RR, A, 0) < C /\ if(B e. RR, B, 0) < D) -> (if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (C + D)) <-> ((if(A e. RR, A, 0) < if(C e. RR, C, 0) /\ if(B e. RR, B, 0) < D) -> (if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (if(C e. RR, C, 0) + D))))
16 breq2 3342 . . . 4 |- (D = if(D e. RR, D, 0) -> (if(B e. RR, B, 0) < D <-> if(B e. RR, B, 0) < if(D e. RR, D, 0)))
1716anbi2d 678 . . 3 |- (D = if(D e. RR, D, 0) -> ((if(A e. RR, A, 0) < if(C e. RR, C, 0) /\ if(B e. RR, B, 0) < D) <-> (if(A e. RR, A, 0) < if(C e. RR, C, 0) /\ if(B e. RR, B, 0) < if(D e. RR, D, 0))))
18 opreq2 4890 . . . 4 |- (D = if(D e. RR, D, 0) -> (if(C e. RR, C, 0) + D) = (if(C e. RR, C, 0) + if(D e. RR, D, 0)))
1918breq2d 3350 . . 3 |- (D = if(D e. RR, D, 0) -> ((if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (if(C e. RR, C, 0) + D) <-> (if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (if(C e. RR, C, 0) + if(D e. RR, D, 0))))
2017, 19imbi12d 688 . 2 |- (D = if(D e. RR, D, 0) -> (((if(A e. RR, A, 0) < if(C e. RR, C, 0) /\ if(B e. RR, B, 0) < D) -> (if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (if(C e. RR, C, 0) + D)) <-> ((if(A e. RR, A, 0) < if(C e. RR, C, 0) /\ if(B e. RR, B, 0) < if(D e. RR, D, 0)) -> (if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (if(C e. RR, C, 0) + if(D e. RR, D, 0)))))
21 0re 6603 . . . 4 |- 0 e. RR
2221elimel 3025 . . 3 |- if(A e. RR, A, 0) e. RR
2321elimel 3025 . . 3 |- if(B e. RR, B, 0) e. RR
2421elimel 3025 . . 3 |- if(C e. RR, C, 0) e. RR
2521elimel 3025 . . 3 |- if(D e. RR, D, 0) e. RR
2622, 23, 24, 25lt2addi 6773 . 2 |- ((if(A e. RR, A, 0) < if(C e. RR, C, 0) /\ if(B e. RR, B, 0) < if(D e. RR, D, 0)) -> (if(A e. RR, A, 0) + if(B e. RR, B, 0)) < (if(C e. RR, C, 0) + if(D e. RR, D, 0)))
275, 10, 15, 20, 26dedth4h 3017 1 |- (((A e. RR /\ B e. RR) /\ (C e. RR /\ D e. RR)) -> ((A < C /\ B < D) -> (A + B) < (C + D)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  ifcif 2982   class class class wbr 3338  (class class class)co 4884  RRcr 6385  0cc0 6386   + caddc 6389   < clt 6653
This theorem is referenced by:  addgt0 6831  nnleltp1 7138  lt2halves 7228  2climnn 8362  2climnn0 8363  climaddlem3 8376  climmullem5 8384  climcaui 8416  bl2in 9120  lmuni 9229  lmcau 9274  vacnlem3 9669  minveclem26 9915  minveclem27 9916  rddif 15798  fsumlt 15821  blhalf 15846  heiborlem16 15970  heiborlem32 15986  heiborlem36 15990
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658
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