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Mirrors > Home > ILE Home > Th. List > 1lt2nq | GIF version |
Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Ref | Expression |
---|---|
1lt2nq | ⊢ 1Q <Q (1Q +Q 1Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2pi 7116 | . . . . 5 ⊢ 1o <N (1o +N 1o) | |
2 | 1pi 7091 | . . . . . 6 ⊢ 1o ∈ N | |
3 | mulidpi 7094 | . . . . . 6 ⊢ (1o ∈ N → (1o ·N 1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (1o ·N 1o) = 1o |
5 | 4, 4 | oveq12i 5754 | . . . . 5 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o) |
6 | 1, 4, 5 | 3brtr4i 3928 | . . . 4 ⊢ (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) |
7 | mulclpi 7104 | . . . . . 6 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o ·N 1o) ∈ N) | |
8 | 2, 2, 7 | mp2an 422 | . . . . 5 ⊢ (1o ·N 1o) ∈ N |
9 | addclpi 7103 | . . . . . 6 ⊢ (((1o ·N 1o) ∈ N ∧ (1o ·N 1o) ∈ N) → ((1o ·N 1o) +N (1o ·N 1o)) ∈ N) | |
10 | 8, 8, 9 | mp2an 422 | . . . . 5 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) ∈ N |
11 | ltmpig 7115 | . . . . 5 ⊢ (((1o ·N 1o) ∈ N ∧ ((1o ·N 1o) +N (1o ·N 1o)) ∈ N ∧ 1o ∈ N) → ((1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))))) | |
12 | 8, 10, 2, 11 | mp3an 1300 | . . . 4 ⊢ ((1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))) |
13 | 6, 12 | mpbi 144 | . . 3 ⊢ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))) |
14 | ordpipqqs 7150 | . . . 4 ⊢ (((1o ∈ N ∧ 1o ∈ N) ∧ (((1o ·N 1o) +N (1o ·N 1o)) ∈ N ∧ (1o ·N 1o) ∈ N)) → ([〈1o, 1o〉] ~Q <Q [〈((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))))) | |
15 | 2, 2, 10, 8, 14 | mp4an 423 | . . 3 ⊢ ([〈1o, 1o〉] ~Q <Q [〈((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q ↔ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o)))) |
16 | 13, 15 | mpbir 145 | . 2 ⊢ [〈1o, 1o〉] ~Q <Q [〈((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q |
17 | df-1nqqs 7127 | . 2 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
18 | 17, 17 | oveq12i 5754 | . . 3 ⊢ (1Q +Q 1Q) = ([〈1o, 1o〉] ~Q +Q [〈1o, 1o〉] ~Q ) |
19 | addpipqqs 7146 | . . . 4 ⊢ (((1o ∈ N ∧ 1o ∈ N) ∧ (1o ∈ N ∧ 1o ∈ N)) → ([〈1o, 1o〉] ~Q +Q [〈1o, 1o〉] ~Q ) = [〈((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q ) | |
20 | 2, 2, 2, 2, 19 | mp4an 423 | . . 3 ⊢ ([〈1o, 1o〉] ~Q +Q [〈1o, 1o〉] ~Q ) = [〈((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q |
21 | 18, 20 | eqtri 2138 | . 2 ⊢ (1Q +Q 1Q) = [〈((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉] ~Q |
22 | 16, 17, 21 | 3brtr4i 3928 | 1 ⊢ 1Q <Q (1Q +Q 1Q) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1316 ∈ wcel 1465 〈cop 3500 class class class wbr 3899 (class class class)co 5742 1oc1o 6274 [cec 6395 Ncnpi 7048 +N cpli 7049 ·N cmi 7050 <N clti 7051 ~Q ceq 7055 1Qc1q 7057 +Q cplq 7058 <Q cltq 7061 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-eprel 4181 df-id 4185 df-iord 4258 df-on 4260 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-irdg 6235 df-1o 6281 df-oadd 6285 df-omul 6286 df-er 6397 df-ec 6399 df-qs 6403 df-ni 7080 df-pli 7081 df-mi 7082 df-lti 7083 df-plpq 7120 df-enq 7123 df-nqqs 7124 df-plqqs 7125 df-1nqqs 7127 df-ltnqqs 7129 |
This theorem is referenced by: ltaddnq 7183 |
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