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Theorem ltaddnq 11014
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 7438 . . 3 (𝑥 = 𝐴 → (𝑥 +Q 𝑦) = (𝐴 +Q 𝑦))
31, 2breq12d 5156 . 2 (𝑥 = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝑦)))
4 oveq2 7439 . . 3 (𝑦 = 𝐵 → (𝐴 +Q 𝑦) = (𝐴 +Q 𝐵))
54breq2d 5155 . 2 (𝑦 = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝐵)))
6 1lt2nq 11013 . . . . . . . 8 1Q <Q (1Q +Q 1Q)
7 ltmnq 11012 . . . . . . . 8 (𝑦Q → (1Q <Q (1Q +Q 1Q) ↔ (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q))))
86, 7mpbii 233 . . . . . . 7 (𝑦Q → (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q)))
9 mulidnq 11003 . . . . . . 7 (𝑦Q → (𝑦 ·Q 1Q) = 𝑦)
10 distrnq 11001 . . . . . . . 8 (𝑦 ·Q (1Q +Q 1Q)) = ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q))
119, 9oveq12d 7449 . . . . . . . 8 (𝑦Q → ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) = (𝑦 +Q 𝑦))
1210, 11eqtrid 2789 . . . . . . 7 (𝑦Q → (𝑦 ·Q (1Q +Q 1Q)) = (𝑦 +Q 𝑦))
138, 9, 123brtr3d 5174 . . . . . 6 (𝑦Q𝑦 <Q (𝑦 +Q 𝑦))
14 ltanq 11011 . . . . . 6 (𝑥Q → (𝑦 <Q (𝑦 +Q 𝑦) ↔ (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
1513, 14imbitrid 244 . . . . 5 (𝑥Q → (𝑦Q → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
1615imp 406 . . . 4 ((𝑥Q𝑦Q) → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))
17 addcomnq 10991 . . . 4 (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥)
18 vex 3484 . . . . 5 𝑥 ∈ V
19 vex 3484 . . . . 5 𝑦 ∈ V
20 addcomnq 10991 . . . . 5 (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)
21 addassnq 10998 . . . . 5 ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡))
2218, 19, 19, 20, 21caov12 7661 . . . 4 (𝑥 +Q (𝑦 +Q 𝑦)) = (𝑦 +Q (𝑥 +Q 𝑦))
2316, 17, 223brtr3g 5176 . . 3 ((𝑥Q𝑦Q) → (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))
24 ltanq 11011 . . . 4 (𝑦Q → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
2524adantl 481 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
2623, 25mpbird 257 . 2 ((𝑥Q𝑦Q) → 𝑥 <Q (𝑥 +Q 𝑦))
273, 5, 26vtocl2ga 3578 1 ((𝐴Q𝐵Q) → 𝐴 <Q (𝐴 +Q 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108   class class class wbr 5143  (class class class)co 7431  Qcnq 10892  1Qc1q 10893   +Q cplq 10895   ·Q cmq 10896   <Q cltq 10898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-ni 10912  df-pli 10913  df-mi 10914  df-lti 10915  df-plpq 10948  df-mpq 10949  df-ltpq 10950  df-enq 10951  df-nq 10952  df-erq 10953  df-plq 10954  df-mq 10955  df-1nq 10956  df-ltnq 10958
This theorem is referenced by:  ltexnq  11015  nsmallnq  11017  ltbtwnnq  11018  prlem934  11073  ltaddpr  11074  ltexprlem2  11077  ltexprlem4  11079
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