Theorem ltaddnq 9834

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	⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 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.

Step	Hyp	Ref	Expression
1		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
2		oveq1 6697	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +Q 𝑦) = (𝐴 +Q 𝑦))
3	1, 2	breq12d 4698	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝑦)))
4		oveq2 6698	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +Q 𝑦) = (𝐴 +Q 𝐵))
5	4	breq2d 4697	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝐵)))
6		1lt2nq 9833	. . . . . . . 8 ⊢ 1Q <Q (1Q +Q 1Q)
7		ltmnq 9832	. . . . . . . 8 ⊢ (𝑦 ∈ Q → (1Q <Q (1Q +Q 1Q) ↔ (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q))))
8	6, 7	mpbii 223	. . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q)))
9		mulidnq 9823	. . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦)
10		distrnq 9821	. . . . . . . 8 ⊢ (𝑦 ·Q (1Q +Q 1Q)) = ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q))
11	9, 9	oveq12d 6708	. . . . . . . 8 ⊢ (𝑦 ∈ Q → ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) = (𝑦 +Q 𝑦))
12	10, 11	syl5eq 2697	. . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (1Q +Q 1Q)) = (𝑦 +Q 𝑦))
13	8, 9, 12	3brtr3d 4716	. . . . . 6 ⊢ (𝑦 ∈ Q → 𝑦 <Q (𝑦 +Q 𝑦))
14		ltanq 9831	. . . . . 6 ⊢ (𝑥 ∈ Q → (𝑦 <Q (𝑦 +Q 𝑦) ↔ (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
15	13, 14	syl5ib 234	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑦 ∈ Q → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
16	15	imp 444	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))
17		addcomnq 9811	. . . 4 ⊢ (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥)
18		vex 3234	. . . . 5 ⊢ 𝑥 ∈ V
19		vex 3234	. . . . 5 ⊢ 𝑦 ∈ V
20		addcomnq 9811	. . . . 5 ⊢ (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)
21		addassnq 9818	. . . . 5 ⊢ ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡))
22	18, 19, 19, 20, 21	caov12 6904	. . . 4 ⊢ (𝑥 +Q (𝑦 +Q 𝑦)) = (𝑦 +Q (𝑥 +Q 𝑦))
23	16, 17, 22	3brtr3g 4718	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))
24		ltanq 9831	. . . 4 ⊢ (𝑦 ∈ Q → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
25	24	adantl 481	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
26	23, 25	mpbird 247	. 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑦))
27	3, 5, 26	vtocl2ga 3305	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   class class class wbr 4685  (class class class)co 6690  Qcnq 9712  1_Qc1q 9713   +_Q cplq 9715   ·_Q cmq 9716   <_Q cltq 9718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-ltnq 9778

This theorem is referenced by:  ltexnq  9835  nsmallnq  9837  ltbtwnnq  9838  prlem934  9893  ltaddpr  9894  ltexprlem2  9897  ltexprlem4  9899