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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 9606 | . . . . . 6 ⊢ 1𝑜 <N (1𝑜 +N 1𝑜) | |
2 | 1pi 9584 | . . . . . . 7 ⊢ 1𝑜 ∈ N | |
3 | mulidpi 9587 | . . . . . . 7 ⊢ (1𝑜 ∈ N → (1𝑜 ·N 1𝑜) = 1𝑜) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (1𝑜 ·N 1𝑜) = 1𝑜 |
5 | addclpi 9593 | . . . . . . . 8 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) ∈ N) | |
6 | 2, 2, 5 | mp2an 704 | . . . . . . 7 ⊢ (1𝑜 +N 1𝑜) ∈ N |
7 | mulidpi 9587 | . . . . . . 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 4613 | . . . . 5 ⊢ (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜) |
10 | ordpipq 9643 | . . . . 5 ⊢ (〈1𝑜, 1𝑜〉 <pQ 〈(1𝑜 +N 1𝑜), 1𝑜〉 ↔ (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜)) | |
11 | 9, 10 | mpbir 220 | . . . 4 ⊢ 〈1𝑜, 1𝑜〉 <pQ 〈(1𝑜 +N 1𝑜), 1𝑜〉 |
12 | df-1nq 9617 | . . . 4 ⊢ 1Q = 〈1𝑜, 1𝑜〉 | |
13 | 12, 12 | oveq12i 6561 | . . . . 5 ⊢ (1Q +pQ 1Q) = (〈1𝑜, 1𝑜〉 +pQ 〈1𝑜, 1𝑜〉) |
14 | addpipq 9638 | . . . . . 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 705 | . . . . 5 ⊢ (〈1𝑜, 1𝑜〉 +pQ 〈1𝑜, 1𝑜〉) = 〈((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)〉 |
16 | 4, 4 | oveq12i 6561 | . . . . . 6 ⊢ ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = (1𝑜 +N 1𝑜) |
17 | 16, 4 | opeq12i 4345 | . . . . 5 ⊢ 〈((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)〉 = 〈(1𝑜 +N 1𝑜), 1𝑜〉 |
18 | 13, 15, 17 | 3eqtri 2636 | . . . 4 ⊢ (1Q +pQ 1Q) = 〈(1𝑜 +N 1𝑜), 1𝑜〉 |
19 | 11, 12, 18 | 3brtr4i 4613 | . . 3 ⊢ 1Q <pQ (1Q +pQ 1Q) |
20 | lterpq 9671 | . . 3 ⊢ (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))) | |
21 | 19, 20 | mpbi 219 | . 2 ⊢ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)) |
22 | 1nq 9629 | . . . 4 ⊢ 1Q ∈ Q | |
23 | nqerid 9634 | . . . 4 ⊢ (1Q ∈ Q → ([Q]‘1Q) = 1Q) | |
24 | 22, 23 | ax-mp 5 | . . 3 ⊢ ([Q]‘1Q) = 1Q |
25 | 24 | eqcomi 2619 | . 2 ⊢ 1Q = ([Q]‘1Q) |
26 | addpqnq 9639 | . . 3 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))) | |
27 | 22, 22, 26 | mp2an 704 | . 2 ⊢ (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)) |
28 | 21, 25, 27 | 3brtr4i 4613 | 1 ⊢ 1Q <Q (1Q +Q 1Q) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 1𝑜c1o 7440 Ncnpi 9545 +N cpli 9546 ·N cmi 9547 <N clti 9548 +pQ cplpq 9549 <pQ cltpq 9551 Qcnq 9553 1Qc1q 9554 [Q]cerq 9555 +Q cplq 9556 <Q cltq 9559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-omul 7452 df-er 7629 df-ni 9573 df-pli 9574 df-mi 9575 df-lti 9576 df-plpq 9609 df-ltpq 9611 df-enq 9612 df-nq 9613 df-erq 9614 df-plq 9615 df-1nq 9617 df-ltnq 9619 |
This theorem is referenced by: ltaddnq 9675 |
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