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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 7341 | . . . . 5 ⊢ 1o <N (1o +N 1o) | |
2 | 1pi 7316 | . . . . . 6 ⊢ 1o ∈ N | |
3 | mulidpi 7319 | . . . . . 6 ⊢ (1o ∈ N → (1o ·N 1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (1o ·N 1o) = 1o |
5 | 4, 4 | oveq12i 5889 | . . . . 5 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o) |
6 | 1, 4, 5 | 3brtr4i 4035 | . . . 4 ⊢ (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) |
7 | mulclpi 7329 | . . . . . 6 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o ·N 1o) ∈ N) | |
8 | 2, 2, 7 | mp2an 426 | . . . . 5 ⊢ (1o ·N 1o) ∈ N |
9 | addclpi 7328 | . . . . . 6 ⊢ (((1o ·N 1o) ∈ N ∧ (1o ·N 1o) ∈ N) → ((1o ·N 1o) +N (1o ·N 1o)) ∈ N) | |
10 | 8, 8, 9 | mp2an 426 | . . . . 5 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) ∈ N |
11 | ltmpig 7340 | . . . . 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 1337 | . . . 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 145 | . . 3 ⊢ (1o ·N (1o ·N 1o)) <N (1o ·N ((1o ·N 1o) +N (1o ·N 1o))) |
14 | ordpipqqs 7375 | . . . 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 427 | . . 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 146 | . 2 ⊢ [⟨1o, 1o⟩] ~Q <Q [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q |
17 | df-1nqqs 7352 | . 2 ⊢ 1Q = [⟨1o, 1o⟩] ~Q | |
18 | 17, 17 | oveq12i 5889 | . . 3 ⊢ (1Q +Q 1Q) = ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) |
19 | addpipqqs 7371 | . . . 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 427 | . . 3 ⊢ ([⟨1o, 1o⟩] ~Q +Q [⟨1o, 1o⟩] ~Q ) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q |
21 | 18, 20 | eqtri 2198 | . 2 ⊢ (1Q +Q 1Q) = [⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩] ~Q |
22 | 16, 17, 21 | 3brtr4i 4035 | 1 ⊢ 1Q <Q (1Q +Q 1Q) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ∈ wcel 2148 ⟨cop 3597 class class class wbr 4005 (class class class)co 5877 1oc1o 6412 [cec 6535 Ncnpi 7273 +N cpli 7274 ·N cmi 7275 <N clti 7276 ~Q ceq 7280 1Qc1q 7282 +Q cplq 7283 <Q cltq 7286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-iinf 4589 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-tr 4104 df-eprel 4291 df-id 4295 df-iord 4368 df-on 4370 df-suc 4373 df-iom 4592 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-ov 5880 df-oprab 5881 df-mpo 5882 df-1st 6143 df-2nd 6144 df-recs 6308 df-irdg 6373 df-1o 6419 df-oadd 6423 df-omul 6424 df-er 6537 df-ec 6539 df-qs 6543 df-ni 7305 df-pli 7306 df-mi 7307 df-lti 7308 df-plpq 7345 df-enq 7348 df-nqqs 7349 df-plqqs 7350 df-1nqqs 7352 df-ltnqqs 7354 |
This theorem is referenced by: ltaddnq 7408 |
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