Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 7214 | . . . . . . 7 ⊢ 1Q <Q (1Q +Q 1Q) | |
2 | 1nq 7174 | . . . . . . . 8 ⊢ 1Q ∈ Q | |
3 | addclnq 7183 | . . . . . . . . 9 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) ∈ Q) | |
4 | 2, 2, 3 | mp2an 422 | . . . . . . . 8 ⊢ (1Q +Q 1Q) ∈ Q |
5 | ltmnqg 7209 | . . . . . . . 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 1305 | . . . . . . 7 ⊢ (𝐵 ∈ Q → (1Q <Q (1Q +Q 1Q) ↔ (𝐵 ·Q 1Q) <Q (𝐵 ·Q (1Q +Q 1Q)))) |
7 | 1, 6 | mpbii 147 | . . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q 1Q) <Q (𝐵 ·Q (1Q +Q 1Q))) |
8 | mulidnq 7197 | . . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q 1Q) = 𝐵) | |
9 | distrnqg 7195 | . . . . . . . 8 ⊢ ((𝐵 ∈ Q ∧ 1Q ∈ Q ∧ 1Q ∈ Q) → (𝐵 ·Q (1Q +Q 1Q)) = ((𝐵 ·Q 1Q) +Q (𝐵 ·Q 1Q))) | |
10 | 2, 2, 9 | mp3an23 1307 | . . . . . . 7 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (1Q +Q 1Q)) = ((𝐵 ·Q 1Q) +Q (𝐵 ·Q 1Q))) |
11 | 8, 8 | oveq12d 5792 | . . . . . . 7 ⊢ (𝐵 ∈ Q → ((𝐵 ·Q 1Q) +Q (𝐵 ·Q 1Q)) = (𝐵 +Q 𝐵)) |
12 | 10, 11 | eqtrd 2172 | . . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 ·Q (1Q +Q 1Q)) = (𝐵 +Q 𝐵)) |
13 | 7, 8, 12 | 3brtr3d 3959 | . . . . 5 ⊢ (𝐵 ∈ Q → 𝐵 <Q (𝐵 +Q 𝐵)) |
14 | 13 | adantl 275 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 <Q (𝐵 +Q 𝐵)) |
15 | simpr 109 | . . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐵 ∈ Q) | |
16 | addclnq 7183 | . . . . . . 7 ⊢ ((𝐵 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 +Q 𝐵) ∈ Q) | |
17 | 16 | anidms 394 | . . . . . 6 ⊢ (𝐵 ∈ Q → (𝐵 +Q 𝐵) ∈ Q) |
18 | 17 | adantl 275 | . . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 +Q 𝐵) ∈ Q) |
19 | simpl 108 | . . . . 5 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 ∈ Q) | |
20 | ltanqg 7208 | . . . . 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 1216 | . . . 4 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 <Q (𝐵 +Q 𝐵) ↔ (𝐴 +Q 𝐵) <Q (𝐴 +Q (𝐵 +Q 𝐵)))) |
22 | 14, 21 | mpbid 146 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) <Q (𝐴 +Q (𝐵 +Q 𝐵))) |
23 | addcomnqg 7189 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) = (𝐵 +Q 𝐴)) | |
24 | addcomnqg 7189 | . . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q) → (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)) | |
25 | 24 | adantl 275 | . . . 4 ⊢ (((𝐴 ∈ Q ∧ 𝐵 ∈ Q) ∧ (𝑟 ∈ Q ∧ 𝑠 ∈ Q)) → (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟)) |
26 | addassnqg 7190 | . . . . 5 ⊢ ((𝑟 ∈ Q ∧ 𝑠 ∈ Q ∧ 𝑡 ∈ Q) → ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡))) | |
27 | 26 | adantl 275 | . . . 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 5952 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q (𝐵 +Q 𝐵)) = (𝐵 +Q (𝐴 +Q 𝐵))) |
29 | 22, 23, 28 | 3brtr3d 3959 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐵 +Q 𝐴) <Q (𝐵 +Q (𝐴 +Q 𝐵))) |
30 | addclnq 7183 | . . 3 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 +Q 𝐵) ∈ Q) | |
31 | ltanqg 7208 | . . 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 1216 | . 2 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → (𝐴 <Q (𝐴 +Q 𝐵) ↔ (𝐵 +Q 𝐴) <Q (𝐵 +Q (𝐴 +Q 𝐵)))) |
33 | 29, 32 | mpbird 166 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 Qcnq 7088 1Qc1q 7089 +Q cplq 7090 ·Q cmq 7091 <Q cltq 7093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-ltnqqs 7161 |
This theorem is referenced by: ltexnqq 7216 nsmallnqq 7220 subhalfnqq 7222 ltbtwnnqq 7223 prarloclemarch2 7227 ltexprlemm 7408 ltexprlemopl 7409 addcanprleml 7422 addcanprlemu 7423 recexprlemm 7432 cauappcvgprlemm 7453 cauappcvgprlemopl 7454 cauappcvgprlem2 7468 caucvgprlemnkj 7474 caucvgprlemnbj 7475 caucvgprlemm 7476 caucvgprlemopl 7477 caucvgprprlemnjltk 7499 caucvgprprlemopl 7505 suplocexprlemmu 7526 |
Copyright terms: Public domain | W3C validator |