Theorem appdiv0nq 7884

Description: Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7883 in which 𝐴 is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)

Assertion

Ref	Expression
appdiv0nq	⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝑚 ·Q 𝐶) <Q 𝐵)

Distinct variable groups:   𝐵,𝑚   𝐶,𝑚

Proof of Theorem appdiv0nq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nsmallnqq 7732	. . 3 ⊢ (𝐵 ∈ Q → ∃𝑥 ∈ Q 𝑥 <Q 𝐵)
2	1	adantr 276	. 2 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑥 ∈ Q 𝑥 <Q 𝐵)
3		appdivnq 7883	. . . . 5 ⊢ ((𝑥 <Q 𝐵 ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝑥 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵))
4		simpr 110	. . . . . 6 ⊢ ((𝑥 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵) → (𝑚 ·Q 𝐶) <Q 𝐵)
5	4	reximi 2641	. . . . 5 ⊢ (∃𝑚 ∈ Q (𝑥 <Q (𝑚 ·Q 𝐶) ∧ (𝑚 ·Q 𝐶) <Q 𝐵) → ∃𝑚 ∈ Q (𝑚 ·Q 𝐶) <Q 𝐵)
6	3, 5	syl 14	. . . 4 ⊢ ((𝑥 <Q 𝐵 ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝑚 ·Q 𝐶) <Q 𝐵)
7	6	ancoms 268	. . 3 ⊢ ((𝐶 ∈ Q ∧ 𝑥 <Q 𝐵) → ∃𝑚 ∈ Q (𝑚 ·Q 𝐶) <Q 𝐵)
8	7	ad2ant2l 508	. 2 ⊢ (((𝐵 ∈ Q ∧ 𝐶 ∈ Q) ∧ (𝑥 ∈ Q ∧ 𝑥 <Q 𝐵)) → ∃𝑚 ∈ Q (𝑚 ·Q 𝐶) <Q 𝐵)
9	2, 8	rexlimddv 2667	1 ⊢ ((𝐵 ∈ Q ∧ 𝐶 ∈ Q) → ∃𝑚 ∈ Q (𝑚 ·Q 𝐶) <Q 𝐵)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2205  ∃wrex 2523   class class class wbr 4111  (class class class)co 6052  Qcnq 7600   ·_Q cmq 7603   <_Q cltq 7605

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-eprel 4412  df-id 4416  df-po 4419  df-iso 4420  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-1o 6649  df-oadd 6653  df-omul 6654  df-er 6769  df-ec 6771  df-qs 6775  df-ni 7624  df-pli 7625  df-mi 7626  df-lti 7627  df-plpq 7664  df-mpq 7665  df-enq 7667  df-nqqs 7668  df-plqqs 7669  df-mqqs 7670  df-1nqqs 7671  df-rq 7672  df-ltnqqs 7673

This theorem is referenced by:  prmuloc  7886