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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 109 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐴 ∈ N) | |
2 | 1pi 7377 | . . . 4 ⊢ 1o ∈ N | |
3 | 2 | a1i 9 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 1o ∈ N) |
4 | simpr 110 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → 𝐵 ∈ N) | |
5 | ordpipqqs 7436 | . . 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 1250 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ([〈𝐴, 1o〉] ~Q <Q [〈𝐵, 1o〉] ~Q ↔ (𝐴 ·N 1o) <N (1o ·N 𝐵))) |
7 | mulidpi 7380 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
8 | 1, 7 | syl 14 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 1o) = 𝐴) |
9 | mulcompig 7393 | . . . . 5 ⊢ ((1o ∈ N ∧ 𝐵 ∈ N) → (1o ·N 𝐵) = (𝐵 ·N 1o)) | |
10 | 2, 4, 9 | sylancr 414 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1o ·N 𝐵) = (𝐵 ·N 1o)) |
11 | mulidpi 7380 | . . . . 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 2226 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1o ·N 𝐵) = 𝐵) |
14 | 8, 13 | breq12d 4043 | . 2 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → ((𝐴 ·N 1o) <N (1o ·N 𝐵) ↔ 𝐴 <N 𝐵)) |
15 | 6, 14 | bitr2d 189 | 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 104 ↔ wb 105 = wceq 1364 ∈ wcel 2164 〈cop 3622 class class class wbr 4030 (class class class)co 5919 1oc1o 6464 [cec 6587 Ncnpi 7334 ·N cmi 7336 <N clti 7337 ~Q ceq 7341 <Q cltq 7347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-eprel 4321 df-id 4325 df-iord 4398 df-on 4400 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-irdg 6425 df-1o 6471 df-oadd 6475 df-omul 6476 df-er 6589 df-ec 6591 df-qs 6595 df-ni 7366 df-mi 7368 df-lti 7369 df-enq 7409 df-nqqs 7410 df-ltnqqs 7415 |
This theorem is referenced by: caucvgprlemk 7727 caucvgprprlemk 7745 ltrennb 7916 |
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