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Mirrors > Home > MPE Home > Th. List > 1lt2nq | Structured version Visualization version GIF version |
Description: One is less than two (one plus one). (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1lt2nq | ⊢ 1Q <Q (1Q +Q 1Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2pi 10321 | . . . . . 6 ⊢ 1o <N (1o +N 1o) | |
2 | 1pi 10299 | . . . . . . 7 ⊢ 1o ∈ N | |
3 | mulidpi 10302 | . . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (1o ·N 1o) = 1o |
5 | addclpi 10308 | . . . . . . . 8 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N) | |
6 | 2, 2, 5 | mp2an 690 | . . . . . . 7 ⊢ (1o +N 1o) ∈ N |
7 | mulidpi 10302 | . . . . . . 7 ⊢ ((1o +N 1o) ∈ N → ((1o +N 1o) ·N 1o) = (1o +N 1o)) | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 ⊢ ((1o +N 1o) ·N 1o) = (1o +N 1o) |
9 | 1, 4, 8 | 3brtr4i 5088 | . . . . 5 ⊢ (1o ·N 1o) <N ((1o +N 1o) ·N 1o) |
10 | ordpipq 10358 | . . . . 5 ⊢ (〈1o, 1o〉 <pQ 〈(1o +N 1o), 1o〉 ↔ (1o ·N 1o) <N ((1o +N 1o) ·N 1o)) | |
11 | 9, 10 | mpbir 233 | . . . 4 ⊢ 〈1o, 1o〉 <pQ 〈(1o +N 1o), 1o〉 |
12 | df-1nq 10332 | . . . 4 ⊢ 1Q = 〈1o, 1o〉 | |
13 | 12, 12 | oveq12i 7162 | . . . . 5 ⊢ (1Q +pQ 1Q) = (〈1o, 1o〉 +pQ 〈1o, 1o〉) |
14 | addpipq 10353 | . . . . . 6 ⊢ (((1o ∈ N ∧ 1o ∈ N) ∧ (1o ∈ N ∧ 1o ∈ N)) → (〈1o, 1o〉 +pQ 〈1o, 1o〉) = 〈((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉) | |
15 | 2, 2, 2, 2, 14 | mp4an 691 | . . . . 5 ⊢ (〈1o, 1o〉 +pQ 〈1o, 1o〉) = 〈((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉 |
16 | 4, 4 | oveq12i 7162 | . . . . . 6 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o) |
17 | 16, 4 | opeq12i 4801 | . . . . 5 ⊢ 〈((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)〉 = 〈(1o +N 1o), 1o〉 |
18 | 13, 15, 17 | 3eqtri 2848 | . . . 4 ⊢ (1Q +pQ 1Q) = 〈(1o +N 1o), 1o〉 |
19 | 11, 12, 18 | 3brtr4i 5088 | . . 3 ⊢ 1Q <pQ (1Q +pQ 1Q) |
20 | lterpq 10386 | . . 3 ⊢ (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))) | |
21 | 19, 20 | mpbi 232 | . 2 ⊢ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)) |
22 | 1nq 10344 | . . . 4 ⊢ 1Q ∈ Q | |
23 | nqerid 10349 | . . . 4 ⊢ (1Q ∈ Q → ([Q]‘1Q) = 1Q) | |
24 | 22, 23 | ax-mp 5 | . . 3 ⊢ ([Q]‘1Q) = 1Q |
25 | 24 | eqcomi 2830 | . 2 ⊢ 1Q = ([Q]‘1Q) |
26 | addpqnq 10354 | . . 3 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))) | |
27 | 22, 22, 26 | mp2an 690 | . 2 ⊢ (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)) |
28 | 21, 25, 27 | 3brtr4i 5088 | 1 ⊢ 1Q <Q (1Q +Q 1Q) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 〈cop 4566 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 1oc1o 8089 Ncnpi 10260 +N cpli 10261 ·N cmi 10262 <N clti 10263 +pQ cplpq 10264 <pQ cltpq 10266 Qcnq 10268 1Qc1q 10269 [Q]cerq 10270 +Q cplq 10271 <Q cltq 10274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-1nq 10332 df-ltnq 10334 |
This theorem is referenced by: ltaddnq 10390 |
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