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Theorem recexpr 6967
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 3811 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑧 <Q 𝑤𝑢 <Q 𝑣))
2 simpr 108 . . . . . . . . 9 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑤 = 𝑣)
32fveq2d 5235 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → (*Q𝑤) = (*Q𝑣))
43eleq1d 2151 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (2nd𝐴) ↔ (*Q𝑣) ∈ (2nd𝐴)))
51, 4anbi12d 457 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ (𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
65cbvexdva 1847 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
76cbvabv 2206 . . . 4 {𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))} = {𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))}
8 simpl 107 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑧 = 𝑢)
92, 8breq12d 3819 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑤 <Q 𝑧𝑣 <Q 𝑢))
103eleq1d 2151 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (1st𝐴) ↔ (*Q𝑣) ∈ (1st𝐴)))
119, 10anbi12d 457 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ (𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1211cbvexdva 1847 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1312cbvabv 2206 . . . 4 {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))} = {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))}
147, 13opeq12i 3596 . . 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 6961 . 2 (𝐴P → ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ ∈ P)
1614recexprlemex 6966 . 2 (𝐴P → (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩) = 1P)
17 oveq2 5573 . . . 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 2091 . . 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 2711 . 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 403 1 (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  {cab 2069  wrex 2354  cop 3420   class class class wbr 3806  cfv 4953  (class class class)co 5565  1st c1st 5818  2nd c2nd 5819  *Qcrq 6613   <Q cltq 6614  Pcnp 6620  1Pc1p 6621   ·P cmp 6623
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-nul 3925  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309  ax-iinf 4358
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-int 3658  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-eprel 4073  df-id 4077  df-po 4080  df-iso 4081  df-iord 4150  df-on 4152  df-suc 4155  df-iom 4361  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-ov 5568  df-oprab 5569  df-mpt2 5570  df-1st 5820  df-2nd 5821  df-recs 5976  df-irdg 6041  df-1o 6087  df-2o 6088  df-oadd 6091  df-omul 6092  df-er 6195  df-ec 6197  df-qs 6201  df-ni 6633  df-pli 6634  df-mi 6635  df-lti 6636  df-plpq 6673  df-mpq 6674  df-enq 6676  df-nqqs 6677  df-plqqs 6678  df-mqqs 6679  df-1nqqs 6680  df-rq 6681  df-ltnqqs 6682  df-enq0 6753  df-nq0 6754  df-0nq0 6755  df-plq0 6756  df-mq0 6757  df-inp 6795  df-i1p 6796  df-imp 6798
This theorem is referenced by:  ltmprr  6971  recexgt0sr  7089
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