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Mirrors > Home > MPE Home > Th. List > ltaddnq | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ltaddnq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
2 | oveq1 6931 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +Q 𝑦) = (𝐴 +Q 𝑦)) | |
3 | 1, 2 | breq12d 4901 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝑦))) |
4 | oveq2 6932 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +Q 𝑦) = (𝐴 +Q 𝐵)) | |
5 | 4 | breq2d 4900 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝐵))) |
6 | 1lt2nq 10132 | . . . . . . . 8 ⊢ 1Q <Q (1Q +Q 1Q) | |
7 | ltmnq 10131 | . . . . . . . 8 ⊢ (𝑦 ∈ Q → (1Q <Q (1Q +Q 1Q) ↔ (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q)))) | |
8 | 6, 7 | mpbii 225 | . . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q))) |
9 | mulidnq 10122 | . . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦) | |
10 | distrnq 10120 | . . . . . . . 8 ⊢ (𝑦 ·Q (1Q +Q 1Q)) = ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) | |
11 | 9, 9 | oveq12d 6942 | . . . . . . . 8 ⊢ (𝑦 ∈ Q → ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) = (𝑦 +Q 𝑦)) |
12 | 10, 11 | syl5eq 2826 | . . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (1Q +Q 1Q)) = (𝑦 +Q 𝑦)) |
13 | 8, 9, 12 | 3brtr3d 4919 | . . . . . 6 ⊢ (𝑦 ∈ Q → 𝑦 <Q (𝑦 +Q 𝑦)) |
14 | ltanq 10130 | . . . . . 6 ⊢ (𝑥 ∈ Q → (𝑦 <Q (𝑦 +Q 𝑦) ↔ (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))) | |
15 | 13, 14 | syl5ib 236 | . . . . 5 ⊢ (𝑥 ∈ Q → (𝑦 ∈ Q → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))) |
16 | 15 | imp 397 | . . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))) |
17 | addcomnq 10110 | . . . 4 ⊢ (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥) | |
18 | vex 3401 | . . . . 5 ⊢ 𝑥 ∈ V | |
19 | vex 3401 | . . . . 5 ⊢ 𝑦 ∈ V | |
20 | addcomnq 10110 | . . . . 5 ⊢ (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟) | |
21 | addassnq 10117 | . . . . 5 ⊢ ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡)) | |
22 | 18, 19, 19, 20, 21 | caov12 7141 | . . . 4 ⊢ (𝑥 +Q (𝑦 +Q 𝑦)) = (𝑦 +Q (𝑥 +Q 𝑦)) |
23 | 16, 17, 22 | 3brtr3g 4921 | . . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))) |
24 | ltanq 10130 | . . . 4 ⊢ (𝑦 ∈ Q → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))) | |
25 | 24 | adantl 475 | . . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))) |
26 | 23, 25 | mpbird 249 | . 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑦)) |
27 | 3, 5, 26 | vtocl2ga 3476 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 class class class wbr 4888 (class class class)co 6924 Qcnq 10011 1Qc1q 10012 +Q cplq 10014 ·Q cmq 10015 <Q cltq 10017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-omul 7850 df-er 8028 df-ni 10031 df-pli 10032 df-mi 10033 df-lti 10034 df-plpq 10067 df-mpq 10068 df-ltpq 10069 df-enq 10070 df-nq 10071 df-erq 10072 df-plq 10073 df-mq 10074 df-1nq 10075 df-ltnq 10077 |
This theorem is referenced by: ltexnq 10134 nsmallnq 10136 ltbtwnnq 10137 prlem934 10192 ltaddpr 10193 ltexprlem2 10196 ltexprlem4 10198 |
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