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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 7338 | . . . . 5 ⊢ 1o <N (1o +N 1o) | |
2 | 1pi 7313 | . . . . . 6 ⊢ 1o ∈ N | |
3 | mulidpi 7316 | . . . . . 6 ⊢ (1o ∈ N → (1o ·N 1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . . 5 ⊢ (1o ·N 1o) = 1o |
5 | 4, 4 | oveq12i 5886 | . . . . 5 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o) |
6 | 1, 4, 5 | 3brtr4i 4033 | . . . 4 ⊢ (1o ·N 1o) <N ((1o ·N 1o) +N (1o ·N 1o)) |
7 | mulclpi 7326 | . . . . . 6 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o ·N 1o) ∈ N) | |
8 | 2, 2, 7 | mp2an 426 | . . . . 5 ⊢ (1o ·N 1o) ∈ N |
9 | addclpi 7325 | . . . . . 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 7337 | . . . . 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 7372 | . . . 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 7349 | . 2 ⊢ 1Q = [〈1o, 1o〉] ~Q | |
18 | 17, 17 | oveq12i 5886 | . . 3 ⊢ (1Q +Q 1Q) = ([〈1o, 1o〉] ~Q +Q [〈1o, 1o〉] ~Q ) |
19 | addpipqqs 7368 | . . . 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 4033 | 1 ⊢ 1Q <Q (1Q +Q 1Q) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1353 ∈ wcel 2148 〈cop 3595 class class class wbr 4003 (class class class)co 5874 1oc1o 6409 [cec 6532 Ncnpi 7270 +N cpli 7271 ·N cmi 7272 <N clti 7273 ~Q ceq 7277 1Qc1q 7279 +Q cplq 7280 <Q cltq 7283 |
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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
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 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-eprel 4289 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-irdg 6370 df-1o 6416 df-oadd 6420 df-omul 6421 df-er 6534 df-ec 6536 df-qs 6540 df-ni 7302 df-pli 7303 df-mi 7304 df-lti 7305 df-plpq 7342 df-enq 7345 df-nqqs 7346 df-plqqs 7347 df-1nqqs 7349 df-ltnqqs 7351 |
This theorem is referenced by: ltaddnq 7405 |
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