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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 108 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐴 ∈ N) | |
2 | 1pi 7264 | . . . 4 ⊢ 1o ∈ N | |
3 | 2 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 1o ∈ N) |
4 | simpr 109 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐵 ∈ N) | |
5 | ordpipqqs 7323 | . . 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 1234 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ([〈𝐴, 1o〉] ~Q <Q [〈𝐵, 1o〉] ~Q ↔ (𝐴 ·N 1o) <N (1o ·N 𝐵))) |
7 | mulidpi 7267 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
8 | 1, 7 | syl 14 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 1o) = 𝐴) |
9 | mulcompig 7280 | . . . . 5 ⊢ ((1o ∈ N ∧ 𝐵 ∈ N) → (1o ·N 𝐵) = (𝐵 ·N 1o)) | |
10 | 2, 4, 9 | sylancr 412 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1o ·N 𝐵) = (𝐵 ·N 1o)) |
11 | mulidpi 7267 | . . . . 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 2203 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1o ·N 𝐵) = 𝐵) |
14 | 8, 13 | breq12d 4000 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 1o) <N (1o ·N 𝐵) ↔ 𝐴 <N 𝐵)) |
15 | 6, 14 | bitr2d 188 | 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 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 〈cop 3584 class class class wbr 3987 (class class class)co 5850 1oc1o 6385 [cec 6507 Ncnpi 7221 ·N cmi 7223 <N clti 7224 ~Q ceq 7228 <Q cltq 7234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-eprel 4272 df-id 4276 df-iord 4349 df-on 4351 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-irdg 6346 df-1o 6392 df-oadd 6396 df-omul 6397 df-er 6509 df-ec 6511 df-qs 6515 df-ni 7253 df-mi 7255 df-lti 7256 df-enq 7296 df-nqqs 7297 df-ltnqqs 7302 |
This theorem is referenced by: caucvgprlemk 7614 caucvgprprlemk 7632 ltrennb 7803 |
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