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Theorem recexpr 6793
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 3796 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑧 <Q 𝑤𝑢 <Q 𝑣))
2 simpr 107 . . . . . . . . 9 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑤 = 𝑣)
32fveq2d 5209 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → (*Q𝑤) = (*Q𝑣))
43eleq1d 2122 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (2nd𝐴) ↔ (*Q𝑣) ∈ (2nd𝐴)))
51, 4anbi12d 450 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ (𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
65cbvexdva 1820 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
76cbvabv 2177 . . . 4 {𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))} = {𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))}
8 simpl 106 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑧 = 𝑢)
92, 8breq12d 3804 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑤 <Q 𝑧𝑣 <Q 𝑢))
103eleq1d 2122 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (1st𝐴) ↔ (*Q𝑣) ∈ (1st𝐴)))
119, 10anbi12d 450 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ (𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1211cbvexdva 1820 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1312cbvabv 2177 . . . 4 {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))} = {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))}
147, 13opeq12i 3581 . . 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 6787 . 2 (𝐴P → ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ ∈ P)
1614recexprlemex 6792 . 2 (𝐴P → (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩) = 1P)
17 oveq2 5547 . . . 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 2064 . . 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 2673 . 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 397 1 (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wrex 2324  cop 3405   class class class wbr 3791  cfv 4929  (class class class)co 5539  1st c1st 5792  2nd c2nd 5793  *Qcrq 6439   <Q cltq 6440  Pcnp 6446  1Pc1p 6447   ·P cmp 6449
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-po 4060  df-iso 4061  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-2o 6032  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-pli 6460  df-mi 6461  df-lti 6462  df-plpq 6499  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-plqqs 6504  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-enq0 6579  df-nq0 6580  df-0nq0 6581  df-plq0 6582  df-mq0 6583  df-inp 6621  df-i1p 6622  df-imp 6624
This theorem is referenced by:  ltmprr  6797  recexgt0sr  6915
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