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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
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