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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 6982 | . . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <N 𝐾 ↔ [〈𝐽, 1𝑜〉] ~Q <Q [〈𝐾, 1𝑜〉] ~Q )) | |
2 | nnnq 6981 | . . . . 5 ⊢ (𝐽 ∈ N → [〈𝐽, 1𝑜〉] ~Q ∈ Q) | |
3 | 2 | adantr 270 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [〈𝐽, 1𝑜〉] ~Q ∈ Q) |
4 | nnnq 6981 | . . . . 5 ⊢ (𝐾 ∈ N → [〈𝐾, 1𝑜〉] ~Q ∈ Q) | |
5 | 4 | adantl 271 | . . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [〈𝐾, 1𝑜〉] ~Q ∈ Q) |
6 | ltnqpr 7152 | . . . 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 403 | . . 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 7106 | . . . . 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 7106 | . . . . 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 7332 | . . . 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 403 | . . 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 212 | . 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 7376 | . 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 196 | 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 102 ↔ wb 103 ∈ wcel 1438 {cab 2074 〈cop 3449 class class class wbr 3845 (class class class)co 5652 1𝑜c1o 6174 [cec 6290 Ncnpi 6831 <N clti 6834 ~Q ceq 6838 Qcnq 6839 <Q cltq 6844 Pcnp 6850 1Pc1p 6851 +P cpp 6852 <P cltp 6854 ~R cer 6855 0Rc0r 6857 <R cltr 6862 <ℝ cltrr 7354 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-13 1449 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-coll 3954 ax-sep 3957 ax-nul 3965 ax-pow 4009 ax-pr 4036 ax-un 4260 ax-setind 4353 ax-iinf 4403 |
This theorem depends on definitions: df-bi 115 df-dc 781 df-3or 925 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-csb 2934 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-nul 3287 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-iun 3732 df-br 3846 df-opab 3900 df-mpt 3901 df-tr 3937 df-eprel 4116 df-id 4120 df-po 4123 df-iso 4124 df-iord 4193 df-on 4195 df-suc 4198 df-iom 4406 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-f 5019 df-f1 5020 df-fo 5021 df-f1o 5022 df-fv 5023 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-1st 5911 df-2nd 5912 df-recs 6070 df-irdg 6135 df-1o 6181 df-2o 6182 df-oadd 6185 df-omul 6186 df-er 6292 df-ec 6294 df-qs 6298 df-ni 6863 df-pli 6864 df-mi 6865 df-lti 6866 df-plpq 6903 df-mpq 6904 df-enq 6906 df-nqqs 6907 df-plqqs 6908 df-mqqs 6909 df-1nqqs 6910 df-rq 6911 df-ltnqqs 6912 df-enq0 6983 df-nq0 6984 df-0nq0 6985 df-plq0 6986 df-mq0 6987 df-inp 7025 df-i1p 7026 df-iplp 7027 df-iltp 7029 df-enr 7272 df-nr 7273 df-ltr 7276 df-0r 7277 df-r 7360 df-lt 7363 |
This theorem is referenced by: ltrenn 7392 axcaucvglemres 7434 |
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