Theorem ltaddnq 11043

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 7455	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +Q 𝑦) = (𝐴 +Q 𝑦))
3	1, 2	breq12d 5179	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝑦)))
4		oveq2 7456	. . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +Q 𝑦) = (𝐴 +Q 𝐵))
5	4	breq2d 5178	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝐵)))
6		1lt2nq 11042	. . . . . . . 8 ⊢ 1Q <Q (1Q +Q 1Q)
7		ltmnq 11041	. . . . . . . 8 ⊢ (𝑦 ∈ Q → (1Q <Q (1Q +Q 1Q) ↔ (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q))))
8	6, 7	mpbii 233	. . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q)))
9		mulidnq 11032	. . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦)
10		distrnq 11030	. . . . . . . 8 ⊢ (𝑦 ·Q (1Q +Q 1Q)) = ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q))
11	9, 9	oveq12d 7466	. . . . . . . 8 ⊢ (𝑦 ∈ Q → ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) = (𝑦 +Q 𝑦))
12	10, 11	eqtrid 2792	. . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (1Q +Q 1Q)) = (𝑦 +Q 𝑦))
13	8, 9, 12	3brtr3d 5197	. . . . . 6 ⊢ (𝑦 ∈ Q → 𝑦 <Q (𝑦 +Q 𝑦))
14		ltanq 11040	. . . . . 6 ⊢ (𝑥 ∈ Q → (𝑦 <Q (𝑦 +Q 𝑦) ↔ (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
15	13, 14	imbitrid 244	. . . . 5 ⊢ (𝑥 ∈ Q → (𝑦 ∈ Q → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))))
16	15	imp 406	. . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))
17		addcomnq 11020	. . . 4 ⊢ (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥)
18		vex 3492	. . . . 5 ⊢ 𝑥 ∈ V
19		vex 3492	. . . . 5 ⊢ 𝑦 ∈ V
20		addcomnq 11020	. . . . 5 ⊢ (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)
21		addassnq 11027	. . . . 5 ⊢ ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡))
22	18, 19, 19, 20, 21	caov12 7678	. . . 4 ⊢ (𝑥 +Q (𝑦 +Q 𝑦)) = (𝑦 +Q (𝑥 +Q 𝑦))
23	16, 17, 22	3brtr3g 5199	. . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))
24		ltanq 11040	. . . 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 257	. 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑦))
27	3, 5, 26	vtocl2ga 3590	1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  (class class class)co 7448  Qcnq 10921  1_Qc1q 10922   +_Q cplq 10924   ·_Q cmq 10925   <_Q cltq 10927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-oadd 8526  df-omul 8527  df-er 8763  df-ni 10941  df-pli 10942  df-mi 10943  df-lti 10944  df-plpq 10977  df-mpq 10978  df-ltpq 10979  df-enq 10980  df-nq 10981  df-erq 10982  df-plq 10983  df-mq 10984  df-1nq 10985  df-ltnq 10987

This theorem is referenced by:  ltexnq  11044  nsmallnq  11046  ltbtwnnq  11047  prlem934  11102  ltaddpr  11103  ltexprlem2  11106  ltexprlem4  11108