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Theorem ltaddnq 10394
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 7156 . . 3 (𝑥 = 𝐴 → (𝑥 +Q 𝑦) = (𝐴 +Q 𝑦))
31, 2breq12d 5065 . 2 (𝑥 = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝑦)))
4 oveq2 7157 . . 3 (𝑦 = 𝐵 → (𝐴 +Q 𝑦) = (𝐴 +Q 𝐵))
54breq2d 5064 . 2 (𝑦 = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝐵)))
6 1lt2nq 10393 . . . . . . . 8 1Q <Q (1Q +Q 1Q)
7 ltmnq 10392 . . . . . . . 8 (𝑦Q → (1Q <Q (1Q +Q 1Q) ↔ (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q))))
86, 7mpbii 236 . . . . . . 7 (𝑦Q → (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q)))
9 mulidnq 10383 . . . . . . 7 (𝑦Q → (𝑦 ·Q 1Q) = 𝑦)
10 distrnq 10381 . . . . . . . 8 (𝑦 ·Q (1Q +Q 1Q)) = ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q))
119, 9oveq12d 7167 . . . . . . . 8 (𝑦Q → ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) = (𝑦 +Q 𝑦))
1210, 11syl5eq 2871 . . . . . . 7 (𝑦Q → (𝑦 ·Q (1Q +Q 1Q)) = (𝑦 +Q 𝑦))
138, 9, 123brtr3d 5083 . . . . . 6 (𝑦Q𝑦 <Q (𝑦 +Q 𝑦))
14 ltanq 10391 . . . . . 6 (𝑥Q → (𝑦 <Q (𝑦 +Q 𝑦) ↔ (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
1513, 14syl5ib 247 . . . . 5 (𝑥Q → (𝑦Q → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
1615imp 410 . . . 4 ((𝑥Q𝑦Q) → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))
17 addcomnq 10371 . . . 4 (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥)
18 vex 3483 . . . . 5 𝑥 ∈ V
19 vex 3483 . . . . 5 𝑦 ∈ V
20 addcomnq 10371 . . . . 5 (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)
21 addassnq 10378 . . . . 5 ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡))
2218, 19, 19, 20, 21caov12 7370 . . . 4 (𝑥 +Q (𝑦 +Q 𝑦)) = (𝑦 +Q (𝑥 +Q 𝑦))
2316, 17, 223brtr3g 5085 . . 3 ((𝑥Q𝑦Q) → (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))
24 ltanq 10391 . . . 4 (𝑦Q → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
2524adantl 485 . . 3 ((𝑥Q𝑦Q) → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
2623, 25mpbird 260 . 2 ((𝑥Q𝑦Q) → 𝑥 <Q (𝑥 +Q 𝑦))
273, 5, 26vtocl2ga 3561 1 ((𝐴Q𝐵Q) → 𝐴 <Q (𝐴 +Q 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115   class class class wbr 5052  (class class class)co 7149  Qcnq 10272  1Qc1q 10273   +Q cplq 10275   ·Q cmq 10276   <Q cltq 10278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-omul 8103  df-er 8285  df-ni 10292  df-pli 10293  df-mi 10294  df-lti 10295  df-plpq 10328  df-mpq 10329  df-ltpq 10330  df-enq 10331  df-nq 10332  df-erq 10333  df-plq 10334  df-mq 10335  df-1nq 10336  df-ltnq 10338
This theorem is referenced by:  ltexnq  10395  nsmallnq  10397  ltbtwnnq  10398  prlem934  10453  ltaddpr  10454  ltexprlem2  10457  ltexprlem4  10459
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