![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ltrennb | GIF version |
Description: Ordering of natural numbers with <N or <ℝ. (Contributed by Jim Kingdon, 13-Jul-2021.) |
Ref | Expression |
---|---|
ltrennb | ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 <ℝ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltnnnq 7440 | . . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ [〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q )) | |
2 | nnnq 7439 | . . . . 5 ⊢ (𝐽 ∈ N → [〈𝐽, 1o〉] ~Q ∈ Q) | |
3 | 2 | adantr 276 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [〈𝐽, 1o〉] ~Q ∈ Q) |
4 | nnnq 7439 | . . . . 5 ⊢ (𝐾 ∈ N → [〈𝐾, 1o〉] ~Q ∈ Q) | |
5 | 4 | adantl 277 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [〈𝐾, 1o〉] ~Q ∈ Q) |
6 | ltnqpr 7610 | . . . 4 ⊢ (([〈𝐽, 1o〉] ~Q ∈ Q ∧ [〈𝐾, 1o〉] ~Q ∈ Q) → ([〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q ↔ 〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉)) | |
7 | 3, 5, 6 | syl2anc 411 | . . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → ([〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q ↔ 〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉)) |
8 | nqprlu 7564 | . . . . 5 ⊢ ([〈𝐽, 1o〉] ~Q ∈ Q → 〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 ∈ P) | |
9 | 3, 8 | syl 14 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → 〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 ∈ P) |
10 | nqprlu 7564 | . . . . 5 ⊢ ([〈𝐾, 1o〉] ~Q ∈ Q → 〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 ∈ P) | |
11 | 5, 10 | syl 14 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → 〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 ∈ P) |
12 | prsrlt 7804 | . . . 4 ⊢ ((〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 ∈ P ∧ 〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 ∈ P) → (〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 ↔ [〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R <R [〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R )) | |
13 | 9, 11, 12 | syl2anc 411 | . . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉<P 〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 ↔ [〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R <R [〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R )) |
14 | 1, 7, 13 | 3bitrd 214 | . 2 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ [〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R <R [〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R )) |
15 | ltresr 7856 | . 2 ⊢ (〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 <ℝ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 ↔ [〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R <R [〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R ) | |
16 | 14, 15 | bitr4di 198 | 1 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐽, 1o〉] ~Q }, {𝑢 ∣ [〈𝐽, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉 <ℝ 〈[〈(〈{𝑙 ∣ 𝑙 <Q [〈𝐾, 1o〉] ~Q }, {𝑢 ∣ [〈𝐾, 1o〉] ~Q <Q 𝑢}〉 +P 1P), 1P〉] ~R , 0R〉)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2160 {cab 2175 〈cop 3610 class class class wbr 4018 (class class class)co 5891 1oc1o 6428 [cec 6551 Ncnpi 7289 <N clti 7292 ~Q ceq 7296 Qcnq 7297 <Q cltq 7302 Pcnp 7308 1Pc1p 7309 +P cpp 7310 <P cltp 7312 ~R cer 7313 0Rc0r 7315 <R cltr 7320 <ℝ cltrr 7833 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
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 4304 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-irdg 6389 df-1o 6435 df-2o 6436 df-oadd 6439 df-omul 6440 df-er 6553 df-ec 6555 df-qs 6559 df-ni 7321 df-pli 7322 df-mi 7323 df-lti 7324 df-plpq 7361 df-mpq 7362 df-enq 7364 df-nqqs 7365 df-plqqs 7366 df-mqqs 7367 df-1nqqs 7368 df-rq 7369 df-ltnqqs 7370 df-enq0 7441 df-nq0 7442 df-0nq0 7443 df-plq0 7444 df-mq0 7445 df-inp 7483 df-i1p 7484 df-iplp 7485 df-iltp 7487 df-enr 7743 df-nr 7744 df-ltr 7747 df-0r 7748 df-r 7839 df-lt 7842 |
This theorem is referenced by: ltrenn 7872 axcaucvglemres 7916 |
Copyright terms: Public domain | W3C validator |