Theorem ltnqpri 7807

Description: We can order fractions via <_Q or <_P. (Contributed by Jim Kingdon, 8-Jan-2021.)

Assertion

Ref	Expression
ltnqpri	⊢ (𝐴 <Q 𝐵 → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)

Distinct variable groups:   𝐴,𝑙   𝑢,𝐴   𝐵,𝑙   𝑢,𝐵

Proof of Theorem ltnqpri

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelnq 7578	. . . . . . . 8 ⊢ <Q ⊆ (Q × Q)
2	1	brel 4776	. . . . . . 7 ⊢ (𝐴 <Q 𝐵 → (𝐴 ∈ Q ∧ 𝐵 ∈ Q))
3	2	simpld 112	. . . . . 6 ⊢ (𝐴 <Q 𝐵 → 𝐴 ∈ Q)
4		nqprlu 7760	. . . . . 6 ⊢ (𝐴 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
5	3, 4	syl 14	. . . . 5 ⊢ (𝐴 <Q 𝐵 → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P)
6	2	simprd 114	. . . . . 6 ⊢ (𝐴 <Q 𝐵 → 𝐵 ∈ Q)
7		nqprlu 7760	. . . . . 6 ⊢ (𝐵 ∈ Q → ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P)
8	6, 7	syl 14	. . . . 5 ⊢ (𝐴 <Q 𝐵 → ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P)
9		ltdfpr 7719	. . . . 5 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩ ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))))
10	5, 8, 9	syl2anc 411	. . . 4 ⊢ (𝐴 <Q 𝐵 → (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩))))
11		vex 2803	. . . . . . 7 ⊢ 𝑥 ∈ V
12		breq2 4090	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝐴 <Q 𝑢 ↔ 𝐴 <Q 𝑥))
13		ltnqex 7762	. . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐴} ∈ V
14		gtnqex 7763	. . . . . . . 8 ⊢ {𝑢 ∣ 𝐴 <Q 𝑢} ∈ V
15	13, 14	op2nd 6305	. . . . . . 7 ⊢ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) = {𝑢 ∣ 𝐴 <Q 𝑢}
16	11, 12, 15	elab2 2952	. . . . . 6 ⊢ (𝑥 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ↔ 𝐴 <Q 𝑥)
17		breq1 4089	. . . . . . 7 ⊢ (𝑙 = 𝑥 → (𝑙 <Q 𝐵 ↔ 𝑥 <Q 𝐵))
18		ltnqex 7762	. . . . . . . 8 ⊢ {𝑙 ∣ 𝑙 <Q 𝐵} ∈ V
19		gtnqex 7763	. . . . . . . 8 ⊢ {𝑢 ∣ 𝐵 <Q 𝑢} ∈ V
20	18, 19	op1st 6304	. . . . . . 7 ⊢ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) = {𝑙 ∣ 𝑙 <Q 𝐵}
21	11, 17, 20	elab2 2952	. . . . . 6 ⊢ (𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩) ↔ 𝑥 <Q 𝐵)
22	16, 21	anbi12i 460	. . . . 5 ⊢ ((𝑥 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ↔ (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵))
23	22	rexbii 2537	. . . 4 ⊢ (∃𝑥 ∈ Q (𝑥 ∈ (2nd ‘⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩) ∧ 𝑥 ∈ (1st ‘⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)) ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵))
24	10, 23	bitrdi 196	. . 3 ⊢ (𝐴 <Q 𝐵 → (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵)))
25		ltbtwnnqq 7628	. . 3 ⊢ (𝐴 <Q 𝐵 ↔ ∃𝑥 ∈ Q (𝐴 <Q 𝑥 ∧ 𝑥 <Q 𝐵))
26	24, 25	bitr4di 198	. 2 ⊢ (𝐴 <Q 𝐵 → (⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩ ↔ 𝐴 <Q 𝐵))
27	26	ibir 177	1 ⊢ (𝐴 <Q 𝐵 → ⟨{𝑙 ∣ 𝑙 <Q 𝐴}, {𝑢 ∣ 𝐴 <Q 𝑢}⟩<P ⟨{𝑙 ∣ 𝑙 <Q 𝐵}, {𝑢 ∣ 𝐵 <Q 𝑢}⟩)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∈ wcel 2200  {cab 2215  ∃wrex 2509  ⟨cop 3670   class class class wbr 4086  ‘cfv 5324  1^st c1st 6296  2^nd c2nd 6297  Qcnq 7493   <_Q cltq 7498  Pcnp 7504  <_P cltp 7508

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7517  df-pli 7518  df-mi 7519  df-lti 7520  df-plpq 7557  df-mpq 7558  df-enq 7560  df-nqqs 7561  df-plqqs 7562  df-mqqs 7563  df-1nqqs 7564  df-rq 7565  df-ltnqqs 7566  df-inp 7679  df-iltp 7683

This theorem is referenced by:  caucvgprprlemk  7896  caucvgprprlemloccalc  7897  caucvgprprlemnjltk  7904  caucvgprprlemlol  7911  caucvgprprlemupu  7913  suplocexprlemloc  7934