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Mirrors > Home > ILE Home > Th. List > ltaddnq | 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.) |
Ref | Expression |
---|---|
ltaddnq | ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2nq 6504 | . . . . . . 7 ⊢ 1Q <Q (1Q +Q 1Q) | |
2 | 1nq 6464 | . . . . . . . 8 ⊢ 1Q ∈ Q | |
3 | addclnq 6473 | . . . . . . . . 9 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) ∈ Q) | |
4 | 2, 2, 3 | mp2an 402 | . . . . . . . 8 ⊢ (1Q +Q 1Q) ∈ Q |
5 | ltmnqg 6499 | . . . . . . . 8 ⊢ ((1Q ∈ Q ∧ (1Q +Q 1Q) ∈ Q ∧ 𝐵 ∈ Q) → (1Q <Q (1Q +Q 1Q) ↔ (𝐵 ·Q 1Q) <Q (𝐵 ·Q (1Q +Q 1Q)))) | |
6 | 2, 4, 5 | mp3an12 1222 | . . . . . . 7 ⊢ (𝐵 ∈ Q → (1Q <Q (1Q +Q 1Q) ↔ (𝐵 ·Q 1Q) <Q (𝐵 ·Q (1Q +Q 1Q)))) |
7 | 1, 6 | mpbii 136 | . . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q 1Q) <Q (𝐵 ·Q (1Q +Q 1Q))) |
8 | mulidnq 6487 | . . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q 1Q) = 𝐵) | |
9 | distrnqg 6485 | . . . . . . . 8 ⊢ ((𝐵 ∈ Q ∧ 1Q ∈ Q ∧ 1Q ∈ Q) → (𝐵 ·Q (1Q +Q 1Q)) = ((𝐵 ·Q 1Q) +Q (𝐵 ·Q 1Q))) | |
10 | 2, 2, 9 | mp3an23 1224 | . . . . . . 7 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (1Q +Q 1Q)) = ((𝐵 ·Q 1Q) +Q (𝐵 ·Q 1Q))) |
11 | 8, 8 | oveq12d 5530 | . . . . . . 7 ⊢ (𝐵 ∈ Q → ((𝐵 ·Q 1Q) +Q (𝐵 ·Q 1Q)) = (𝐵 +Q 𝐵)) |
12 | 10, 11 | eqtrd 2072 | . . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (1Q +Q 1Q)) = (𝐵 +Q 𝐵)) |
13 | 7, 8, 12 | 3brtr3d 3793 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 <Q (𝐵 +Q 𝐵)) |
14 | 13 | adantl 262 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 <Q (𝐵 +Q 𝐵)) |
15 | simpr 103 | . . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 ∈ Q) | |
16 | addclnq 6473 | . . . . . . 7 ⊢ ((𝐵 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 +Q 𝐵) ∈ Q) | |
17 | 16 | anidms 377 | . . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 +Q 𝐵) ∈ Q) |
18 | 17 | adantl 262 | . . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 +Q 𝐵) ∈ Q) |
19 | simpl 102 | . . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 ∈ Q) | |
20 | ltanqg 6498 | . . . . 5 ⊢ ((𝐵 ∈ Q ∧ (𝐵 +Q 𝐵) ∈ Q ∧ 𝐴 ∈ Q) → (𝐵 <Q (𝐵 +Q 𝐵) ↔ (𝐴 +Q 𝐵) <Q (𝐴 +Q (𝐵 +Q 𝐵)))) | |
21 | 15, 18, 19, 20 | syl3anc 1135 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 <Q (𝐵 +Q 𝐵) ↔ (𝐴 +Q 𝐵) <Q (𝐴 +Q (𝐵 +Q 𝐵)))) |
22 | 14, 21 | mpbid 135 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) <Q (𝐴 +Q (𝐵 +Q 𝐵))) |
23 | addcomnqg 6479 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴)) | |
24 | addcomnqg 6479 | . . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q) → (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)) | |
25 | 24 | adantl 262 | . . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝑟 ∈ Q ∧ 𝑠 ∈ Q)) → (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)) |
26 | addassnqg 6480 | . . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q ∧ 𝑡 ∈ Q) → ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡))) | |
27 | 26 | adantl 262 | . . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝑟 ∈ Q ∧ 𝑠 ∈ Q ∧ 𝑡 ∈ Q)) → ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡))) |
28 | 19, 15, 15, 25, 27 | caov12d 5682 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q (𝐵 +Q 𝐵)) = (𝐵 +Q (𝐴 +Q 𝐵))) |
29 | 22, 23, 28 | 3brtr3d 3793 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 +Q 𝐴) <Q (𝐵 +Q (𝐴 +Q 𝐵))) |
30 | addclnq 6473 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q) | |
31 | ltanqg 6498 | . . 3 ⊢ ((𝐴 ∈ Q ∧ (𝐴 +Q 𝐵) ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q (𝐴 +Q 𝐵) ↔ (𝐵 +Q 𝐴) <Q (𝐵 +Q (𝐴 +Q 𝐵)))) | |
32 | 19, 30, 15, 31 | syl3anc 1135 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q (𝐴 +Q 𝐵) ↔ (𝐵 +Q 𝐴) <Q (𝐵 +Q (𝐴 +Q 𝐵)))) |
33 | 29, 32 | mpbird 156 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 class class class wbr 3764 (class class class)co 5512 Qcnq 6378 1Qc1q 6379 +Q cplq 6380 ·Q cmq 6381 <Q cltq 6383 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-ltnqqs 6451 |
This theorem is referenced by: ltexnqq 6506 nsmallnqq 6510 subhalfnqq 6512 ltbtwnnqq 6513 prarloclemarch2 6517 ltexprlemm 6698 ltexprlemopl 6699 addcanprleml 6712 addcanprlemu 6713 recexprlemm 6722 cauappcvgprlemm 6743 cauappcvgprlemopl 6744 cauappcvgprlem2 6758 caucvgprlemnkj 6764 caucvgprlemnbj 6765 caucvgprlemm 6766 caucvgprlemopl 6767 caucvgprprlemnjltk 6789 caucvgprprlemopl 6795 |
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