Theorem ltaddnq 11064

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 23	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
2		oveq1 7429	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +Q 𝑦) = (𝐴 +Q 𝑦))
3	1, 2	breq12d 5115	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝑦)))
4		oveq2 7430	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +Q 𝑦) = (𝐴 +Q 𝐵))
5	4	breq2d 5114	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝐵)))
6		1lt2nq 11063	. . . . . . . 8 ⊢ 1Q <Q (1Q +Q 1Q)
7		ltmnq 11062	. . . . . . . 8 ⊢ (𝑦 ∈ Q → (1Q <Q (1Q +Q 1Q) ↔ (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q))))
8	6, 7	mpbii 236	. . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q)))
9		mulidnq 11053	. . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦)
10		distrnq 11051	. . . . . . . 8 ⊢ (𝑦 ·Q (1Q +Q 1Q)) = ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q))
11	9, 9	oveq12d 7440	. . . . . . . 8 ⊢ (𝑦 ∈ Q → ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) = (𝑦 +Q 𝑦))
12	10, 11	eqtrid 2807	. . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (1Q +Q 1Q)) = (𝑦 +Q 𝑦))
13	8, 9, 12	3brtr3d 5135	. . . . . 6 ⊢ (𝑦 ∈ Q → 𝑦 <Q (𝑦 +Q 𝑦))
14		ltanq 11061	. . . . . 6 ⊢ (𝑥 ∈ Q → (𝑦 <Q (𝑦 +Q 𝑦) ↔ (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
15	13, 14	imbitrid 247	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑦 ∈ Q → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
16	15	imp 412	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))
17		addcomnq 11041	. . . 4 ⊢ (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥)
18		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
19		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
20		addcomnq 11041	. . . . 5 ⊢ (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)
21		addassnq 11048	. . . . 5 ⊢ ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡))
22	18, 19, 19, 20, 21	caov12 7651	. . . 4 ⊢ (𝑥 +Q (𝑦 +Q 𝑦)) = (𝑦 +Q (𝑥 +Q 𝑦))
23	16, 17, 22	3brtr3g 5137	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))
24		ltanq 11061	. . . 4 ⊢ (𝑦 ∈ Q → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
25	24	adantl 487	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))))
26	23, 25	mpbird 260	. 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑦))
27	3, 5, 26	vtocl2ga 3537	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   class class class wbr 5102  (class class class)co 7422  Qcnq 10942  1_Qc1q 10943   +_Q cplq 10945   ·_Q cmq 10946   <_Q cltq 10948

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-oadd 8485  df-omul 8486  df-er 8722  df-ni 10962  df-pli 10963  df-mi 10964  df-lti 10965  df-plpq 10998  df-mpq 10999  df-ltpq 11000  df-enq 11001  df-nq 11002  df-erq 11003  df-plq 11004  df-mq 11005  df-1nq 11006  df-ltnq 11008

This theorem is used by:  ltexnq  11065  nsmallnq  11067  ltbtwnnq  11068  prlem934  11123  ltaddpr  11124  ltexprlem2  11127  ltexprlem4  11129