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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 10897 | . . . . . 6 ⊢ 1o <N (1o +N 1o) | |
2 | 1pi 10875 | . . . . . . 7 ⊢ 1o ∈ N | |
3 | mulidpi 10878 | . . . . . . 7 ⊢ (1o ∈ N → (1o ·N 1o) = 1o) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (1o ·N 1o) = 1o |
5 | addclpi 10884 | . . . . . . . 8 ⊢ ((1o ∈ N ∧ 1o ∈ N) → (1o +N 1o) ∈ N) | |
6 | 2, 2, 5 | mp2an 691 | . . . . . . 7 ⊢ (1o +N 1o) ∈ N |
7 | mulidpi 10878 | . . . . . . 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 5178 | . . . . 5 ⊢ (1o ·N 1o) <N ((1o +N 1o) ·N 1o) |
10 | ordpipq 10934 | . . . . 5 ⊢ (⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩ ↔ (1o ·N 1o) <N ((1o +N 1o) ·N 1o)) | |
11 | 9, 10 | mpbir 230 | . . . 4 ⊢ ⟨1o, 1o⟩ <pQ ⟨(1o +N 1o), 1o⟩ |
12 | df-1nq 10908 | . . . 4 ⊢ 1Q = ⟨1o, 1o⟩ | |
13 | 12, 12 | oveq12i 7418 | . . . . 5 ⊢ (1Q +pQ 1Q) = (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩) |
14 | addpipq 10929 | . . . . . 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 692 | . . . . 5 ⊢ (⟨1o, 1o⟩ +pQ ⟨1o, 1o⟩) = ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ |
16 | 4, 4 | oveq12i 7418 | . . . . . 6 ⊢ ((1o ·N 1o) +N (1o ·N 1o)) = (1o +N 1o) |
17 | 16, 4 | opeq12i 4878 | . . . . 5 ⊢ ⟨((1o ·N 1o) +N (1o ·N 1o)), (1o ·N 1o)⟩ = ⟨(1o +N 1o), 1o⟩ |
18 | 13, 15, 17 | 3eqtri 2765 | . . . 4 ⊢ (1Q +pQ 1Q) = ⟨(1o +N 1o), 1o⟩ |
19 | 11, 12, 18 | 3brtr4i 5178 | . . 3 ⊢ 1Q <pQ (1Q +pQ 1Q) |
20 | lterpq 10962 | . . 3 ⊢ (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))) | |
21 | 19, 20 | mpbi 229 | . 2 ⊢ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)) |
22 | 1nq 10920 | . . . 4 ⊢ 1Q ∈ Q | |
23 | nqerid 10925 | . . . 4 ⊢ (1Q ∈ Q → ([Q]‘1Q) = 1Q) | |
24 | 22, 23 | ax-mp 5 | . . 3 ⊢ ([Q]‘1Q) = 1Q |
25 | 24 | eqcomi 2742 | . 2 ⊢ 1Q = ([Q]‘1Q) |
26 | addpqnq 10930 | . . 3 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))) | |
27 | 22, 22, 26 | mp2an 691 | . 2 ⊢ (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)) |
28 | 21, 25, 27 | 3brtr4i 5178 | 1 ⊢ 1Q <Q (1Q +Q 1Q) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 ⟨cop 4634 class class class wbr 5148 ‘cfv 6541 (class class class)co 7406 1oc1o 8456 Ncnpi 10836 +N cpli 10837 ·N cmi 10838 <N clti 10839 +pQ cplpq 10840 <pQ cltpq 10842 Qcnq 10844 1Qc1q 10845 [Q]cerq 10846 +Q cplq 10847 <Q cltq 10850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-oadd 8467 df-omul 8468 df-er 8700 df-ni 10864 df-pli 10865 df-mi 10866 df-lti 10867 df-plpq 10900 df-ltpq 10902 df-enq 10903 df-nq 10904 df-erq 10905 df-plq 10906 df-1nq 10908 df-ltnq 10910 |
This theorem is referenced by: ltaddnq 10966 |
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