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Theorem recexpr 7469
Description: The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
Assertion
Ref Expression
recexpr (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Distinct variable group:   𝑥,𝐴

Proof of Theorem recexpr
Dummy variables 𝑢 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 3941 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑧 <Q 𝑤𝑢 <Q 𝑣))
2 simpr 109 . . . . . . . . 9 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑤 = 𝑣)
32fveq2d 5432 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → (*Q𝑤) = (*Q𝑣))
43eleq1d 2209 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (2nd𝐴) ↔ (*Q𝑣) ∈ (2nd𝐴)))
51, 4anbi12d 465 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ (𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
65cbvexdva 1902 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
76cbvabv 2265 . . . 4 {𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))} = {𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))}
8 simpl 108 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑧 = 𝑢)
92, 8breq12d 3949 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑤 <Q 𝑧𝑣 <Q 𝑢))
103eleq1d 2209 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (1st𝐴) ↔ (*Q𝑣) ∈ (1st𝐴)))
119, 10anbi12d 465 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ (𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1211cbvexdva 1902 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1312cbvabv 2265 . . . 4 {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))} = {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))}
147, 13opeq12i 3717 . . 3 ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ = ⟨{𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))}, {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))}⟩
1514recexprlempr 7463 . 2 (𝐴P → ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ ∈ P)
1614recexprlemex 7468 . 2 (𝐴P → (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩) = 1P)
17 oveq2 5789 . . . 4 (𝑥 = ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ → (𝐴 ·P 𝑥) = (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩))
1817eqeq1d 2149 . . 3 (𝑥 = ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ → ((𝐴 ·P 𝑥) = 1P ↔ (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩) = 1P))
1918rspcev 2792 . 2 ((⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ ∈ P ∧ (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩) = 1P) → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
2015, 16, 19syl2anc 409 1 (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wex 1469  wcel 1481  {cab 2126  wrex 2418  cop 3534   class class class wbr 3936  cfv 5130  (class class class)co 5781  1st c1st 6043  2nd c2nd 6044  *Qcrq 7115   <Q cltq 7116  Pcnp 7122  1Pc1p 7123   ·P cmp 7125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-eprel 4218  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-irdg 6274  df-1o 6320  df-2o 6321  df-oadd 6324  df-omul 6325  df-er 6436  df-ec 6438  df-qs 6442  df-ni 7135  df-pli 7136  df-mi 7137  df-lti 7138  df-plpq 7175  df-mpq 7176  df-enq 7178  df-nqqs 7179  df-plqqs 7180  df-mqqs 7181  df-1nqqs 7182  df-rq 7183  df-ltnqqs 7184  df-enq0 7255  df-nq0 7256  df-0nq0 7257  df-plq0 7258  df-mq0 7259  df-inp 7297  df-i1p 7298  df-imp 7300
This theorem is referenced by:  ltmprr  7473  recexgt0sr  7604
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