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Theorem reclem4pr 10084
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 10082 . . . . . 6 (𝐴P𝐵P)
3 df-mp 10018 . . . . . . 7 ·P = (𝑦P, 𝑤P ↦ {𝑢 ∣ ∃𝑓𝑦𝑔𝑤 𝑢 = (𝑓 ·Q 𝑔)})
4 mulclnq 9981 . . . . . . 7 ((𝑓Q𝑔Q) → (𝑓 ·Q 𝑔) ∈ Q)
53, 4genpelv 10034 . . . . . 6 ((𝐴P𝐵P) → (𝑤 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑧𝐴𝑥𝐵 𝑤 = (𝑧 ·Q 𝑥)))
62, 5mpdan 705 . . . . 5 (𝐴P → (𝑤 ∈ (𝐴 ·P 𝐵) ↔ ∃𝑧𝐴𝑥𝐵 𝑤 = (𝑧 ·Q 𝑥)))
71abeq2i 2873 . . . . . . . . 9 (𝑥𝐵 ↔ ∃𝑦(𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴))
8 ltrelnq 9960 . . . . . . . . . . . . . . 15 <Q ⊆ (Q × Q)
98brel 5325 . . . . . . . . . . . . . 14 (𝑥 <Q 𝑦 → (𝑥Q𝑦Q))
109simprd 482 . . . . . . . . . . . . 13 (𝑥 <Q 𝑦𝑦Q)
11 elprnq 10025 . . . . . . . . . . . . . . . . . . 19 ((𝐴P𝑧𝐴) → 𝑧Q)
12 ltmnq 10006 . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . 16 (((𝐴P𝑧𝐴) ∧ 𝑦Q) → (𝑥 <Q 𝑦 → (𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦)))
16 recclnq 10000 . . . . . . . . . . . . . . . . . 18 (𝑦Q → (*Q𝑦) ∈ Q)
17 prub 10028 . . . . . . . . . . . . . . . . . 18 (((𝐴P𝑧𝐴) ∧ (*Q𝑦) ∈ Q) → (¬ (*Q𝑦) ∈ 𝐴𝑧 <Q (*Q𝑦)))
1816, 17sylan2 492 . . . . . . . . . . . . . . . . 17 (((𝐴P𝑧𝐴) ∧ 𝑦Q) → (¬ (*Q𝑦) ∈ 𝐴𝑧 <Q (*Q𝑦)))
19 ltmnq 10006 . . . . . . . . . . . . . . . . . . 19 (𝑦Q → (𝑧 <Q (*Q𝑦) ↔ (𝑦 ·Q 𝑧) <Q (𝑦 ·Q (*Q𝑦))))
20 mulcomnq 9987 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦)
2120a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑦Q → (𝑦 ·Q 𝑧) = (𝑧 ·Q 𝑦))
22 recidnq 9999 . . . . . . . . . . . . . . . . . . . 20 (𝑦Q → (𝑦 ·Q (*Q𝑦)) = 1Q)
2321, 22breq12d 4817 . . . . . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . 15 (((𝐴P𝑧𝐴) ∧ 𝑦Q) → ((𝑥 <Q 𝑦 ∧ ¬ (*Q𝑦) ∈ 𝐴) → ((𝑧 ·Q 𝑥) <Q (𝑧 ·Q 𝑦) ∧ (𝑧 ·Q 𝑦) <Q 1Q)))
28 ltsonq 10003 . . . . . . . . . . . . . . . 16 <Q Or Q
2928, 8sotri 5681 . . . . . . . . . . . . . . 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 633 . . . . . . . . . . . . 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 2010 . . . . . . . . 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 4807 . . . . . . . . 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 630 . . . . . 6 (𝐴P → ((𝑧𝐴𝑥𝐵) → (𝑤 = (𝑧 ·Q 𝑥) → 𝑤 <Q 1Q)))
4140rexlimdvv 3175 . . . . 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 10016 . . . . 5 1P = {𝑤𝑤 <Q 1Q}
4443abeq2i 2873 . . . 4 (𝑤 ∈ 1P𝑤 <Q 1Q)
4542, 44syl6ibr 242 . . 3 (𝐴P → (𝑤 ∈ (𝐴 ·P 𝐵) → 𝑤 ∈ 1P))
4645ssrdv 3750 . 2 (𝐴P → (𝐴 ·P 𝐵) ⊆ 1P)
471reclem3pr 10083 . 2 (𝐴P → 1P ⊆ (𝐴 ·P 𝐵))
4846, 47eqssd 3761 1 (𝐴P → (𝐴 ·P 𝐵) = 1P)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wrex 3051   class class class wbr 4804  cfv 6049  (class class class)co 6814  Qcnq 9886  1Qc1q 9887   ·Q cmq 9890  *Qcrq 9891   <Q cltq 9892  Pcnp 9893  1Pc1p 9894   ·P cmp 9896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-omul 7735  df-er 7913  df-ni 9906  df-pli 9907  df-mi 9908  df-lti 9909  df-plpq 9942  df-mpq 9943  df-ltpq 9944  df-enq 9945  df-nq 9946  df-erq 9947  df-plq 9948  df-mq 9949  df-1nq 9950  df-rq 9951  df-ltnq 9952  df-np 10015  df-1p 10016  df-mp 10018
This theorem is referenced by:  recexpr  10085
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