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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 7163 | . . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +Q 𝑦) = (𝐴 +Q 𝑦)) | |
3 | 1, 2 | breq12d 5079 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 <Q (𝑥 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝑦))) |
4 | oveq2 7164 | . . 3 ⊢ (𝑦 = 𝐵 → (𝐴 +Q 𝑦) = (𝐴 +Q 𝐵)) | |
5 | 4 | breq2d 5078 | . 2 ⊢ (𝑦 = 𝐵 → (𝐴 <Q (𝐴 +Q 𝑦) ↔ 𝐴 <Q (𝐴 +Q 𝐵))) |
6 | 1lt2nq 10395 | . . . . . . . 8 ⊢ 1Q <Q (1Q +Q 1Q) | |
7 | ltmnq 10394 | . . . . . . . 8 ⊢ (𝑦 ∈ Q → (1Q <Q (1Q +Q 1Q) ↔ (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q)))) | |
8 | 6, 7 | mpbii 235 | . . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) <Q (𝑦 ·Q (1Q +Q 1Q))) |
9 | mulidnq 10385 | . . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q 1Q) = 𝑦) | |
10 | distrnq 10383 | . . . . . . . 8 ⊢ (𝑦 ·Q (1Q +Q 1Q)) = ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) | |
11 | 9, 9 | oveq12d 7174 | . . . . . . . 8 ⊢ (𝑦 ∈ Q → ((𝑦 ·Q 1Q) +Q (𝑦 ·Q 1Q)) = (𝑦 +Q 𝑦)) |
12 | 10, 11 | syl5eq 2868 | . . . . . . 7 ⊢ (𝑦 ∈ Q → (𝑦 ·Q (1Q +Q 1Q)) = (𝑦 +Q 𝑦)) |
13 | 8, 9, 12 | 3brtr3d 5097 | . . . . . 6 ⊢ (𝑦 ∈ Q → 𝑦 <Q (𝑦 +Q 𝑦)) |
14 | ltanq 10393 | . . . . . 6 ⊢ (𝑥 ∈ Q → (𝑦 <Q (𝑦 +Q 𝑦) ↔ (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))) | |
15 | 13, 14 | syl5ib 246 | . . . . 5 ⊢ (𝑥 ∈ Q → (𝑦 ∈ Q → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦)))) |
16 | 15 | imp 409 | . . . 4 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 +Q 𝑦) <Q (𝑥 +Q (𝑦 +Q 𝑦))) |
17 | addcomnq 10373 | . . . 4 ⊢ (𝑥 +Q 𝑦) = (𝑦 +Q 𝑥) | |
18 | vex 3497 | . . . . 5 ⊢ 𝑥 ∈ V | |
19 | vex 3497 | . . . . 5 ⊢ 𝑦 ∈ V | |
20 | addcomnq 10373 | . . . . 5 ⊢ (𝑟 +Q 𝑠) = (𝑠 +Q 𝑟) | |
21 | addassnq 10380 | . . . . 5 ⊢ ((𝑟 +Q 𝑠) +Q 𝑡) = (𝑟 +Q (𝑠 +Q 𝑡)) | |
22 | 18, 19, 19, 20, 21 | caov12 7376 | . . . 4 ⊢ (𝑥 +Q (𝑦 +Q 𝑦)) = (𝑦 +Q (𝑥 +Q 𝑦)) |
23 | 16, 17, 22 | 3brtr3g 5099 | . . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦))) |
24 | ltanq 10393 | . . . 4 ⊢ (𝑦 ∈ Q → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))) | |
25 | 24 | adantl 484 | . . 3 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → (𝑥 <Q (𝑥 +Q 𝑦) ↔ (𝑦 +Q 𝑥) <Q (𝑦 +Q (𝑥 +Q 𝑦)))) |
26 | 23, 25 | mpbird 259 | . 2 ⊢ ((𝑥 ∈ Q ∧ 𝑦 ∈ Q) → 𝑥 <Q (𝑥 +Q 𝑦)) |
27 | 3, 5, 26 | vtocl2ga 3575 | 1 ⊢ ((𝐴 ∈ Q ∧ 𝐵 ∈ Q) → 𝐴 <Q (𝐴 +Q 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 (class class class)co 7156 Qcnq 10274 1Qc1q 10275 +Q cplq 10277 ·Q cmq 10278 <Q cltq 10280 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-omul 8107 df-er 8289 df-ni 10294 df-pli 10295 df-mi 10296 df-lti 10297 df-plpq 10330 df-mpq 10331 df-ltpq 10332 df-enq 10333 df-nq 10334 df-erq 10335 df-plq 10336 df-mq 10337 df-1nq 10338 df-ltnq 10340 |
This theorem is referenced by: ltexnq 10397 nsmallnq 10399 ltbtwnnq 10400 prlem934 10455 ltaddpr 10456 ltexprlem2 10459 ltexprlem4 10461 |
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