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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 7231 | . . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ [〈𝐽, 1o〉] ~Q <Q [〈𝐾, 1o〉] ~Q )) | |
2 | nnnq 7230 | . . . . 5 ⊢ (𝐽 ∈ N → [〈𝐽, 1o〉] ~Q ∈ Q) | |
3 | 2 | adantr 274 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [〈𝐽, 1o〉] ~Q ∈ Q) |
4 | nnnq 7230 | . . . . 5 ⊢ (𝐾 ∈ N → [〈𝐾, 1o〉] ~Q ∈ Q) | |
5 | 4 | adantl 275 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [〈𝐾, 1o〉] ~Q ∈ Q) |
6 | ltnqpr 7401 | . . . 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 408 | . . 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 7355 | . . . . 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 7355 | . . . . 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 7595 | . . . 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 408 | . . 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 213 | . 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 7647 | . 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 | syl6bbr 197 | 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 103 ↔ wb 104 ∈ wcel 1480 {cab 2125 〈cop 3530 class class class wbr 3929 (class class class)co 5774 1oc1o 6306 [cec 6427 Ncnpi 7080 <N clti 7083 ~Q ceq 7087 Qcnq 7088 <Q cltq 7093 Pcnp 7099 1Pc1p 7100 +P cpp 7101 <P cltp 7103 ~R cer 7104 0Rc0r 7106 <R cltr 7111 <ℝ cltrr 7624 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-eprel 4211 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-irdg 6267 df-1o 6313 df-2o 6314 df-oadd 6317 df-omul 6318 df-er 6429 df-ec 6431 df-qs 6435 df-ni 7112 df-pli 7113 df-mi 7114 df-lti 7115 df-plpq 7152 df-mpq 7153 df-enq 7155 df-nqqs 7156 df-plqqs 7157 df-mqqs 7158 df-1nqqs 7159 df-rq 7160 df-ltnqqs 7161 df-enq0 7232 df-nq0 7233 df-0nq0 7234 df-plq0 7235 df-mq0 7236 df-inp 7274 df-i1p 7275 df-iplp 7276 df-iltp 7278 df-enr 7534 df-nr 7535 df-ltr 7538 df-0r 7539 df-r 7630 df-lt 7633 |
This theorem is referenced by: ltrenn 7663 axcaucvglemres 7707 |
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