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Theorem recexpr 7636
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 4008 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑧 <Q 𝑤𝑢 <Q 𝑣))
2 simpr 110 . . . . . . . . 9 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑤 = 𝑣)
32fveq2d 5519 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → (*Q𝑤) = (*Q𝑣))
43eleq1d 2246 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (2nd𝐴) ↔ (*Q𝑣) ∈ (2nd𝐴)))
51, 4anbi12d 473 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ (𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
65cbvexdva 1929 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴)) ↔ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))))
76cbvabv 2302 . . . 4 {𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))} = {𝑢 ∣ ∃𝑣(𝑢 <Q 𝑣 ∧ (*Q𝑣) ∈ (2nd𝐴))}
8 simpl 109 . . . . . . . 8 ((𝑧 = 𝑢𝑤 = 𝑣) → 𝑧 = 𝑢)
92, 8breq12d 4016 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → (𝑤 <Q 𝑧𝑣 <Q 𝑢))
103eleq1d 2246 . . . . . . 7 ((𝑧 = 𝑢𝑤 = 𝑣) → ((*Q𝑤) ∈ (1st𝐴) ↔ (*Q𝑣) ∈ (1st𝐴)))
119, 10anbi12d 473 . . . . . 6 ((𝑧 = 𝑢𝑤 = 𝑣) → ((𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ (𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1211cbvexdva 1929 . . . . 5 (𝑧 = 𝑢 → (∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴)) ↔ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))))
1312cbvabv 2302 . . . 4 {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))} = {𝑢 ∣ ∃𝑣(𝑣 <Q 𝑢 ∧ (*Q𝑣) ∈ (1st𝐴))}
147, 13opeq12i 3783 . . 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 7630 . 2 (𝐴P → ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩ ∈ P)
1614recexprlemex 7635 . 2 (𝐴P → (𝐴 ·P ⟨{𝑧 ∣ ∃𝑤(𝑧 <Q 𝑤 ∧ (*Q𝑤) ∈ (2nd𝐴))}, {𝑧 ∣ ∃𝑤(𝑤 <Q 𝑧 ∧ (*Q𝑤) ∈ (1st𝐴))}⟩) = 1P)
17 oveq2 5882 . . . 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 2186 . . 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 2841 . 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 411 1 (𝐴P → ∃𝑥P (𝐴 ·P 𝑥) = 1P)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492  wcel 2148  {cab 2163  wrex 2456  cop 3595   class class class wbr 4003  cfv 5216  (class class class)co 5874  1st c1st 6138  2nd c2nd 6139  *Qcrq 7282   <Q cltq 7283  Pcnp 7289  1Pc1p 7290   ·P cmp 7292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-2o 6417  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-pli 7303  df-mi 7304  df-lti 7305  df-plpq 7342  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-plqqs 7347  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-enq0 7422  df-nq0 7423  df-0nq0 7424  df-plq0 7425  df-mq0 7426  df-inp 7464  df-i1p 7465  df-imp 7467
This theorem is referenced by:  ltmprr  7640  recexgt0sr  7771
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