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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 7345 | . . . 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 7404 | . . 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 7348 | . . . 4 ⊢ (𝐴 ∈ N → (𝐴 ·N 1o) = 𝐴) | |
8 | 1, 7 | syl 14 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (𝐴 ·N 1o) = 𝐴) |
9 | mulcompig 7361 | . . . . 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 7348 | . . . . 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 2222 | . . 3 ⊢ ((𝐴 ∈ N ∧ 𝐵 ∈ N) → (1o ·N 𝐵) = 𝐵) |
14 | 8, 13 | breq12d 4031 | . 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 2160 〈cop 3610 class class class wbr 4018 (class class class)co 5897 1oc1o 6435 [cec 6558 Ncnpi 7302 ·N cmi 7304 <N clti 7305 ~Q ceq 7309 <Q cltq 7315 |
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 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
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 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-eprel 4307 df-id 4311 df-iord 4384 df-on 4386 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-irdg 6396 df-1o 6442 df-oadd 6446 df-omul 6447 df-er 6560 df-ec 6562 df-qs 6566 df-ni 7334 df-mi 7336 df-lti 7337 df-enq 7377 df-nqqs 7378 df-ltnqqs 7383 |
This theorem is referenced by: caucvgprlemk 7695 caucvgprprlemk 7713 ltrennb 7884 |
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