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Theorem reclem4pr 9910
Description: Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (New usage is discouraged.)
Hypothesis
Ref Expression
reclempr.1 𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}
Assertion
Ref Expression
reclem4pr (𝐴P → (𝐴 ·P 𝐵) = 1P)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem reclem4pr
Dummy variables 𝑧 𝑤 𝑢 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reclempr.1 . . . . . . 7 𝐵 = {𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴)}
21reclem2pr 9908 . . . . . 6 (𝐴P𝐵P)
3 df-mp 9844 . . . . . . 7 ·P = (𝑦P, 𝑤P ↦ {𝑢 ∣ ∃𝑓𝑦𝑔𝑤 𝑢 = (𝑓 ·Q 𝑔)})
4 mulclnq 9807 . . . . . . 7 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
53, 4genpelv 9860 . . . . . 6 ((𝐴P𝐵P) → (𝑤 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑧𝐴𝑥𝐵 𝑤 = (𝑧 ·Q 𝑥)))
62, 5mpdan 703 . . . . 5 (𝐴P → (𝑤 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑧𝐴𝑥𝐵 𝑤 = (𝑧 ·Q 𝑥)))
71abeq2i 2764 . . . . . . . . 9 (𝑥𝐵 ↔ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴))
8 ltrelnq 9786 . . . . . . . . . . . . . . 15 <Q ⊆ (Q × Q)
98brel 5202 . . . . . . . . . . . . . 14 (𝑥 <Q 𝑦 → (𝑥Q𝑦Q))
109simprd 478 . . . . . . . . . . . . 13 (𝑥 <Q 𝑦𝑦Q)
11 elprnq 9851 . . . . . . . . . . . . . . . . . . 19 ((𝐴P𝑧𝐴) → 𝑧Q)
12 ltmnq 9832 . . . . . . . . . . . . . . . . . . 19 (𝑧Q → (𝑥 <Q 𝑦 ↔ (𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦)))
1311, 12syl 17 . . . . . . . . . . . . . . . . . 18 ((𝐴P𝑧𝐴) → (𝑥 <Q 𝑦 ↔ (𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦)))
1413biimpd 219 . . . . . . . . . . . . . . . . 17 ((𝐴P𝑧𝐴) → (𝑥 <Q 𝑦 → (𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦)))
1514adantr 480 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧𝐴) ∧ 𝑦Q) → (𝑥 <Q 𝑦 → (𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦)))
16 recclnq 9826 . . . . . . . . . . . . . . . . . 18 (𝑦Q → (*Q𝑦) ∈ Q)
17 prub 9854 . . . . . . . . . . . . . . . . . 18 (((𝐴P𝑧𝐴) ∧ (*Q𝑦) ∈ Q) → (¬ (*Q𝑦) ∈ 𝐴𝑧 <Q (*Q𝑦)))
1816, 17sylan2 490 . . . . . . . . . . . . . . . . 17 (((𝐴P𝑧𝐴) ∧ 𝑦Q) → (¬ (*Q𝑦) ∈ 𝐴𝑧 <Q (*Q𝑦)))
19 ltmnq 9832 . . . . . . . . . . . . . . . . . . 19 (𝑦Q → (𝑧 <Q (*Q𝑦) ↔ (𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦))))
20 mulcomnq 9813 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
2120a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑦Q → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
22 recidnq 9825 . . . . . . . . . . . . . . . . . . . 20 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
2321, 22breq12d 4698 . . . . . . . . . . . . . . . . . . 19 (𝑦Q → ((𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦)) ↔ (𝑧 ·Q 𝑦) <Q 1Q))
2419, 23bitrd 268 . . . . . . . . . . . . . . . . . 18 (𝑦Q → (𝑧 <Q (*Q𝑦) ↔ (𝑧 ·Q 𝑦) <Q 1Q))
2524adantl 481 . . . . . . . . . . . . . . . . 17 (((𝐴P𝑧𝐴) ∧ 𝑦Q) → (𝑧 <Q (*Q𝑦) ↔ (𝑧 ·Q 𝑦) <Q 1Q))
2618, 25sylibd 229 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧𝐴) ∧ 𝑦Q) → (¬ (*Q𝑦) ∈ 𝐴 → (𝑧 ·Q 𝑦) <Q 1Q))
2715, 26anim12d 585 . . . . . . . . . . . . . . 15 (((𝐴P𝑧𝐴) ∧ 𝑦Q) → ((𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) → ((𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q 1Q)))
28 ltsonq 9829 . . . . . . . . . . . . . . . 16 <Q Or Q
2928, 8sotri 5558 . . . . . . . . . . . . . . 15 (((𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q 1Q) → (𝑧 ·Q 𝑥) <Q 1Q)
3027, 29syl6 35 . . . . . . . . . . . . . 14 (((𝐴P𝑧𝐴) ∧ 𝑦Q) → ((𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) → (𝑧 ·Q 𝑥) <Q 1Q))
3130exp4b 631 . . . . . . . . . . . . 13 ((𝐴P𝑧𝐴) → (𝑦Q → (𝑥 <Q 𝑦 → (¬ (*Q𝑦) ∈ 𝐴 → (𝑧 ·Q 𝑥) <Q 1Q))))
3210, 31syl5 34 . . . . . . . . . . . 12 ((𝐴P𝑧𝐴) → (𝑥 <Q 𝑦 → (𝑥 <Q 𝑦 → (¬ (*Q𝑦) ∈ 𝐴 → (𝑧 ·Q 𝑥) <Q 1Q))))
3332pm2.43d 53 . . . . . . . . . . 11 ((𝐴P𝑧𝐴) → (𝑥 <Q 𝑦 → (¬ (*Q𝑦) ∈ 𝐴 → (𝑧 ·Q 𝑥) <Q 1Q)))
3433impd 446 . . . . . . . . . 10 ((𝐴P𝑧𝐴) → ((𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) → (𝑧 ·Q 𝑥) <Q 1Q))
3534exlimdv 1901 . . . . . . . . 9 ((𝐴P𝑧𝐴) → (∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) → (𝑧 ·Q 𝑥) <Q 1Q))
367, 35syl5bi 232 . . . . . . . 8 ((𝐴P𝑧𝐴) → (𝑥𝐵 → (𝑧 ·Q 𝑥) <Q 1Q))
37 breq1 4688 . . . . . . . . 9 (𝑤 = (𝑧 ·Q 𝑥) → (𝑤 <Q 1Q ↔ (𝑧 ·Q 𝑥) <Q 1Q))
3837biimprcd 240 . . . . . . . 8 ((𝑧 ·Q 𝑥) <Q 1Q → (𝑤 = (𝑧 ·Q 𝑥) → 𝑤 <Q 1Q))
3936, 38syl6 35 . . . . . . 7 ((𝐴P𝑧𝐴) → (𝑥𝐵 → (𝑤 = (𝑧 ·Q 𝑥) → 𝑤 <Q 1Q)))
4039expimpd 628 . . . . . 6 (𝐴P → ((𝑧𝐴𝑥𝐵) → (𝑤 = (𝑧 ·Q 𝑥) → 𝑤 <Q 1Q)))
4140rexlimdvv 3066 . . . . 5 (𝐴P → (∃𝑧𝐴𝑥𝐵 𝑤 = (𝑧 ·Q 𝑥) → 𝑤 <Q 1Q))
426, 41sylbid 230 . . . 4 (𝐴P → (𝑤 ∈ (𝐴 ·P 𝐵) → 𝑤 <Q 1Q))
43 df-1p 9842 . . . . 5 1P = {𝑤𝑤 <Q 1Q}
4443abeq2i 2764 . . . 4 (𝑤 ∈ 1P𝑤 <Q 1Q)
4542, 44syl6ibr 242 . . 3 (𝐴P → (𝑤 ∈ (𝐴 ·P 𝐵) → 𝑤 ∈ 1P))
4645ssrdv 3642 . 2 (𝐴P → (𝐴 ·P 𝐵) ⊆ 1P)
471reclem3pr 9909 . 2 (𝐴P → 1P ⊆ (𝐴 ·P 𝐵))
4846, 47eqssd 3653 1 (𝐴P → (𝐴 ·P 𝐵) = 1P)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wrex 2942   class class class wbr 4685  cfv 5926  (class class class)co 6690  Qcnq 9712  1Qc1q 9713   ·Q cmq 9716  *Qcrq 9717   <Q cltq 9718  Pcnp 9719  1Pc1p 9720   ·P cmp 9722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-omul 7610  df-er 7787  df-ni 9732  df-pli 9733  df-mi 9734  df-lti 9735  df-plpq 9768  df-mpq 9769  df-ltpq 9770  df-enq 9771  df-nq 9772  df-erq 9773  df-plq 9774  df-mq 9775  df-1nq 9776  df-rq 9777  df-ltnq 9778  df-np 9841  df-1p 9842  df-mp 9844
This theorem is referenced by:  recexpr  9911
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