ltrennb - Intuitionistic Logic Explorer

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Theorem ltrennb 7022

Description: Ordering of natural numbers with <_N or <_ℝ. (Contributed by Jim Kingdon, 13-Jul-2021.)

**Assertion**
Ref	Expression
ltrennb	⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ <_ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩))

Distinct variable groups: 𝐽,𝑙 𝑢,𝐽 𝐾,𝑙 𝑢,𝐾

**Proof of Theorem ltrennb**
Step	Hyp	Ref	Expression
1		ltnnnq 6613	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q [⟨𝐾, 1_𝑜⟩] ~_Q ))
2		nnnq 6612	. . . . 5 ⊢ (𝐽 ∈ N → [⟨𝐽, 1_𝑜⟩] ~_Q ∈ Q)
3	2	adantr 270	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐽, 1_𝑜⟩] ~_Q ∈ Q)
4		nnnq 6612	. . . . 5 ⊢ (𝐾 ∈ N → [⟨𝐾, 1_𝑜⟩] ~_Q ∈ Q)
5	4	adantl 271	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → [⟨𝐾, 1_𝑜⟩] ~_Q ∈ Q)
6		ltnqpr 6783	. . . 4 ⊢ (([⟨𝐽, 1_𝑜⟩] ~_Q ∈ Q ∧ [⟨𝐾, 1_𝑜⟩] ~_Q ∈ Q) → ([⟨𝐽, 1_𝑜⟩] ~_Q <_Q [⟨𝐾, 1_𝑜⟩] ~_Q ↔ ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩))
7	3, 5, 6	syl2anc 403	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → ([⟨𝐽, 1_𝑜⟩] ~_Q <_Q [⟨𝐾, 1_𝑜⟩] ~_Q ↔ ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩))
8		nqprlu 6737	. . . . 5 ⊢ ([⟨𝐽, 1_𝑜⟩] ~_Q ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ∈ P)
9	3, 8	syl 14	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ∈ P)
10		nqprlu 6737	. . . . 5 ⊢ ([⟨𝐾, 1_𝑜⟩] ~_Q ∈ Q → ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ∈ P)
11	5, 10	syl 14	. . . 4 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ∈ P)
12		prsrlt 6963	. . . 4 ⊢ ((⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ∈ P) → (⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ↔ [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R <_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ))
13	9, 11, 12	syl2anc 403	. . 3 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩<_P ⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ ↔ [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R <_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ))
14	1, 7, 13	3bitrd 212	. 2 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R <_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R ))
15		ltresr 7007	. 2 ⊢ (⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ <_ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ ↔ [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R <_R [⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R )
16	14, 15	syl6bbr 196	1 ⊢ ((𝐽 ∈ N ∧ 𝐾 ∈ N) → (𝐽 <_N 𝐾 ↔ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐽, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐽, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩ <_ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <_Q [⟨𝐾, 1_𝑜⟩] ~_Q }, {𝑢 ∣ [⟨𝐾, 1_𝑜⟩] ~_Q <_Q 𝑢}⟩ +_P 1_P), 1_P⟩] ~_R , 0_R⟩))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∈ wcel 1433 {cab 2067 ⟨cop 3401 class class class wbr 3785 (class class class)co 5532 1_𝑜c1o 6017 [cec 6127 Ncnpi 6462 <_N clti 6465 ~_Q ceq 6469 Qcnq 6470 <_Q cltq 6475 Pcnp 6481 1_Pc1p 6482 +_P cpp 6483 <_P cltp 6485 ~_R cer 6486 0_Rc0r 6488 <_R cltr 6493 <_ℝ cltrr 6985

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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329

This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-i1p 6657 df-iplp 6658 df-iltp 6660 df-enr 6903 df-nr 6904 df-ltr 6907 df-0r 6908 df-r 6991 df-lt 6994

This theorem is referenced by: ltrenn 7023 axcaucvglemres 7065

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