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Mirrors > Home > ILE Home > Th. List > ltnnnq | GIF version |
Description: Ordering of positive integers via <N or <Q is equivalent. (Contributed by Jim Kingdon, 3-Oct-2020.) |
Ref | Expression |
---|---|
ltnnnq | ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ [〈𝐴, 1𝑜〉] ~Q <Q [〈𝐵, 1𝑜〉] ~Q )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐴 ∈ N) | |
2 | 1pi 6874 | . . . 4 ⊢ 1𝑜 ∈ N | |
3 | 2 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 1𝑜 ∈ N) |
4 | simpr 108 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐵 ∈ N) | |
5 | ordpipqqs 6933 | . . 3 ⊢ (((𝐴 ∈ N ∧ 1𝑜 ∈ N) ∧ (𝐵 ∈ N ∧ 1𝑜 ∈ N)) → ([〈𝐴, 1𝑜〉] ~Q <Q [〈𝐵, 1𝑜〉] ~Q ↔ (𝐴 ·N 1𝑜) <N (1𝑜 ·N 𝐵))) | |
6 | 1, 3, 4, 3, 5 | syl22anc 1175 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ([〈𝐴, 1𝑜〉] ~Q <Q [〈𝐵, 1𝑜〉] ~Q ↔ (𝐴 ·N 1𝑜) <N (1𝑜 ·N 𝐵))) |
7 | mulidpi 6877 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 ·N 1𝑜) = 𝐴) | |
8 | 1, 7 | syl 14 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 1𝑜) = 𝐴) |
9 | mulcompig 6890 | . . . . 5 ⊢ ((1𝑜 ∈ N ∧ 𝐵 ∈ N) → (1𝑜 ·N 𝐵) = (𝐵 ·N 1𝑜)) | |
10 | 2, 4, 9 | sylancr 405 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1𝑜 ·N 𝐵) = (𝐵 ·N 1𝑜)) |
11 | mulidpi 6877 | . . . . 5 ⊢ (𝐵 ∈ N → (𝐵 ·N 1𝑜) = 𝐵) | |
12 | 4, 11 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐵 ·N 1𝑜) = 𝐵) |
13 | 10, 12 | eqtrd 2120 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1𝑜 ·N 𝐵) = 𝐵) |
14 | 8, 13 | breq12d 3858 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 1𝑜) <N (1𝑜 ·N 𝐵) ↔ 𝐴 <N 𝐵)) |
15 | 6, 14 | bitr2d 187 | 1 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 <N 𝐵 ↔ [〈𝐴, 1𝑜〉] ~Q <Q [〈𝐵, 1𝑜〉] ~Q )) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1289 ∈ wcel 1438 〈cop 3449 class class class wbr 3845 (class class class)co 5652 1𝑜c1o 6174 [cec 6290 Ncnpi 6831 ·N cmi 6833 <N clti 6834 ~Q ceq 6838 <Q cltq 6844 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-eprel 4116 df-id 4120 df-iord 4193 df-on 4195 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-irdg 6135 df-1o 6181 df-oadd 6185 df-omul 6186 df-er 6292 df-ec 6294 df-qs 6298 df-ni 6863 df-mi 6865 df-lti 6866 df-enq 6906 df-nqqs 6907 df-ltnqqs 6912 |
This theorem is referenced by: caucvgprlemk 7224 caucvgprprlemk 7242 ltrennb 7391 |
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