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Theorem ltaddnq 9756
Description: The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltaddnq ((𝐴Q𝐵Q) → 𝐴 <Q (𝐴 +Q 𝐵))

Proof of Theorem ltaddnq
Dummy variables 𝑥 𝑦 𝑠 𝑟 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝑥 = 𝐴𝑥 = 𝐴)
2 oveq1 6622 . . 3 (𝑥 = 𝐴 → (𝑥 +Q 𝑦) = (𝐴 +Q 𝑦))
31, 2breq12d 4636 . 2 (𝑥 = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝑦)))
4 oveq2 6623 . . 3 (𝑦 = 𝐵 → (𝐴 +Q 𝑦) = (𝐴 +Q 𝐵))
54breq2d 4635 . 2 (𝑦 = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝐵)))
6 1lt2nq 9755 . . . . . . . 8 1Q <Q (1Q +Q 1Q)
7 ltmnq 9754 . . . . . . . 8 (𝑦Q → (1Q <Q (1Q +Q 1Q) ↔ (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q))))
86, 7mpbii 223 . . . . . . 7 (𝑦Q → (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q)))
9 mulidnq 9745 . . . . . . 7 (𝑦Q → (𝑦 ·Q 1Q) = 𝑦)
10 distrnq 9743 . . . . . . . 8 (𝑦 ·Q (1Q +Q 1Q)) = ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q))
119, 9oveq12d 6633 . . . . . . . 8 (𝑦Q → ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) = (𝑦 +Q 𝑦))
1210, 11syl5eq 2667 . . . . . . 7 (𝑦Q → (𝑦 ·Q (1Q +Q 1Q)) = (𝑦 +Q 𝑦))
138, 9, 123brtr3d 4654 . . . . . 6 (𝑦Q𝑦 <Q (𝑦 +Q 𝑦))
14 ltanq 9753 . . . . . 6 (𝑥Q → (𝑦 <Q (𝑦 +Q 𝑦) ↔ (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
1513, 14syl5ib 234 . . . . 5 (𝑥Q → (𝑦Q → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
1615imp 445 . . . 4 ((𝑥Q𝑦Q) → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))
17 addcomnq 9733 . . . 4 (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥)
18 vex 3193 . . . . 5 𝑥 ∈ V
19 vex 3193 . . . . 5 𝑦 ∈ V
20 addcomnq 9733 . . . . 5 (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)
21 addassnq 9740 . . . . 5 ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡))
2218, 19, 19, 20, 21caov12 6827 . . . 4 (𝑥 +Q (𝑦 +Q 𝑦)) = (𝑦 +Q (𝑥 +Q 𝑦))
2316, 17, 223brtr3g 4656 . . 3 ((𝑥Q𝑦Q) → (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))
24 ltanq 9753 . . . 4 (𝑦Q → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
2524adantl 482 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
2623, 25mpbird 247 . 2 ((𝑥Q𝑦Q) → 𝑥 <Q (𝑥 +Q 𝑦))
273, 5, 26vtocl2ga 3264 1 ((𝐴Q𝐵Q) → 𝐴 <Q (𝐴 +Q 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987   class class class wbr 4623  (class class class)co 6615  Qcnq 9634  1Qc1q 9635   +Q cplq 9637   ·Q cmq 9638   <Q cltq 9640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-ni 9654  df-pli 9655  df-mi 9656  df-lti 9657  df-plpq 9690  df-mpq 9691  df-ltpq 9692  df-enq 9693  df-nq 9694  df-erq 9695  df-plq 9696  df-mq 9697  df-1nq 9698  df-ltnq 9700
This theorem is referenced by:  ltexnq  9757  nsmallnq  9759  ltbtwnnq  9760  prlem934  9815  ltaddpr  9816  ltexprlem2  9819  ltexprlem4  9821
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