Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 9727 | . . . . . 6 ⊢ 1𝑜 <N (1𝑜 +N 1𝑜) | |
2 | 1pi 9705 | . . . . . . 7 ⊢ 1𝑜 ∈ N | |
3 | mulidpi 9708 | . . . . . . 7 ⊢ (1𝑜 ∈ N → (1𝑜 ·N 1𝑜) = 1𝑜) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ (1𝑜 ·N 1𝑜) = 1𝑜 |
5 | addclpi 9714 | . . . . . . . 8 ⊢ ((1𝑜 ∈ N ∧ 1𝑜 ∈ N) → (1𝑜 +N 1𝑜) ∈ N) | |
6 | 2, 2, 5 | mp2an 708 | . . . . . . 7 ⊢ (1𝑜 +N 1𝑜) ∈ N |
7 | mulidpi 9708 | . . . . . . 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 4683 | . . . . 5 ⊢ (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜) |
10 | ordpipq 9764 | . . . . 5 ⊢ (〈1𝑜, 1𝑜〉 <pQ 〈(1𝑜 +N 1𝑜), 1𝑜〉 ↔ (1𝑜 ·N 1𝑜) <N ((1𝑜 +N 1𝑜) ·N 1𝑜)) | |
11 | 9, 10 | mpbir 221 | . . . 4 ⊢ 〈1𝑜, 1𝑜〉 <pQ 〈(1𝑜 +N 1𝑜), 1𝑜〉 |
12 | df-1nq 9738 | . . . 4 ⊢ 1Q = 〈1𝑜, 1𝑜〉 | |
13 | 12, 12 | oveq12i 6662 | . . . . 5 ⊢ (1Q +pQ 1Q) = (〈1𝑜, 1𝑜〉 +pQ 〈1𝑜, 1𝑜〉) |
14 | addpipq 9759 | . . . . . 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 709 | . . . . 5 ⊢ (〈1𝑜, 1𝑜〉 +pQ 〈1𝑜, 1𝑜〉) = 〈((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)〉 |
16 | 4, 4 | oveq12i 6662 | . . . . . 6 ⊢ ((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)) = (1𝑜 +N 1𝑜) |
17 | 16, 4 | opeq12i 4407 | . . . . 5 ⊢ 〈((1𝑜 ·N 1𝑜) +N (1𝑜 ·N 1𝑜)), (1𝑜 ·N 1𝑜)〉 = 〈(1𝑜 +N 1𝑜), 1𝑜〉 |
18 | 13, 15, 17 | 3eqtri 2648 | . . . 4 ⊢ (1Q +pQ 1Q) = 〈(1𝑜 +N 1𝑜), 1𝑜〉 |
19 | 11, 12, 18 | 3brtr4i 4683 | . . 3 ⊢ 1Q <pQ (1Q +pQ 1Q) |
20 | lterpq 9792 | . . 3 ⊢ (1Q <pQ (1Q +pQ 1Q) ↔ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q))) | |
21 | 19, 20 | mpbi 220 | . 2 ⊢ ([Q]‘1Q) <Q ([Q]‘(1Q +pQ 1Q)) |
22 | 1nq 9750 | . . . 4 ⊢ 1Q ∈ Q | |
23 | nqerid 9755 | . . . 4 ⊢ (1Q ∈ Q → ([Q]‘1Q) = 1Q) | |
24 | 22, 23 | ax-mp 5 | . . 3 ⊢ ([Q]‘1Q) = 1Q |
25 | 24 | eqcomi 2631 | . 2 ⊢ 1Q = ([Q]‘1Q) |
26 | addpqnq 9760 | . . 3 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q))) | |
27 | 22, 22, 26 | mp2an 708 | . 2 ⊢ (1Q +Q 1Q) = ([Q]‘(1Q +pQ 1Q)) |
28 | 21, 25, 27 | 3brtr4i 4683 | 1 ⊢ 1Q <Q (1Q +Q 1Q) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 1𝑜c1o 7553 Ncnpi 9666 +N cpli 9667 ·N cmi 9668 <N clti 9669 +pQ cplpq 9670 <pQ cltpq 9672 Qcnq 9674 1Qc1q 9675 [Q]cerq 9676 +Q cplq 9677 <Q cltq 9680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-ltpq 9732 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-1nq 9738 df-ltnq 9740 |
This theorem is referenced by: ltaddnq 9796 |
Copyright terms: Public domain | W3C validator |