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Theorem recexprlemex 6678
Description: 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 6679. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
Assertion
Ref Expression
recexprlemex (𝐴P → (𝐴 ·P 𝐵) = 1P)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlemex
StepHypRef Expression
1 recexpr.1 . . . 4 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
21recexprlemss1l 6676 . . 3 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
31recexprlem1ssl 6674 . . 3 (𝐴P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))
42, 3eqssd 2959 . 2 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P))
51recexprlemss1u 6677 . . 3 (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
61recexprlem1ssu 6675 . . 3 (𝐴P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
75, 6eqssd 2959 . 2 (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))
81recexprlempr 6673 . . . 4 (𝐴P𝐵P)
9 mulclpr 6613 . . . 4 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
108, 9mpdan 398 . . 3 (𝐴P → (𝐴 ·P 𝐵) ∈ P)
11 1pr 6595 . . 3 1PP
12 preqlu 6513 . . 3 (((𝐴 ·P 𝐵) ∈ P ∧ 1PP) → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))))
1310, 11, 12sylancl 392 . 2 (𝐴P → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))))
144, 7, 13mpbir2and 851 1 (𝐴P → (𝐴 ·P 𝐵) = 1P)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wb 98   = wceq 1243  wex 1381  wcel 1393  {cab 2026  cop 3375   class class class wbr 3760  cfv 4863  (class class class)co 5473  1st c1st 5726  2nd c2nd 5727  *Qcrq 6325   <Q cltq 6326  Pcnp 6332  1Pc1p 6333   ·P cmp 6335
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3868  ax-sep 3871  ax-nul 3879  ax-pow 3923  ax-pr 3940  ax-un 4141  ax-setind 4230  ax-iinf 4272
This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-int 3612  df-iun 3655  df-br 3761  df-opab 3815  df-mpt 3816  df-tr 3851  df-eprel 4022  df-id 4026  df-po 4029  df-iso 4030  df-iord 4074  df-on 4076  df-suc 4079  df-iom 4275  df-xp 4312  df-rel 4313  df-cnv 4314  df-co 4315  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319  df-iota 4828  df-fun 4865  df-fn 4866  df-f 4867  df-f1 4868  df-fo 4869  df-f1o 4870  df-fv 4871  df-ov 5476  df-oprab 5477  df-mpt2 5478  df-1st 5728  df-2nd 5729  df-recs 5881  df-irdg 5918  df-1o 5962  df-2o 5963  df-oadd 5966  df-omul 5967  df-er 6065  df-ec 6067  df-qs 6071  df-ni 6345  df-pli 6346  df-mi 6347  df-lti 6348  df-plpq 6385  df-mpq 6386  df-enq 6388  df-nqqs 6389  df-plqqs 6390  df-mqqs 6391  df-1nqqs 6392  df-rq 6393  df-ltnqqs 6394  df-enq0 6465  df-nq0 6466  df-0nq0 6467  df-plq0 6468  df-mq0 6469  df-inp 6507  df-i1p 6508  df-imp 6510
This theorem is referenced by:  recexpr  6679
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