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Theorem genpml 7339
Description: The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.)
Hypotheses
Ref Expression
genpelvl.1 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genpml ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑞,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑞   𝑥,𝐺,𝑦,𝑧,𝑤,𝑣,𝑞   𝐹,𝑞
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpml
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 7297 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prml 7299 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑓Q 𝑓 ∈ (1st𝐴))
3 rexex 2479 . . . 4 (∃𝑓Q 𝑓 ∈ (1st𝐴) → ∃𝑓 𝑓 ∈ (1st𝐴))
41, 2, 33syl 17 . . 3 (𝐴P → ∃𝑓 𝑓 ∈ (1st𝐴))
54adantr 274 . 2 ((𝐴P𝐵P) → ∃𝑓 𝑓 ∈ (1st𝐴))
6 prop 7297 . . . . 5 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
7 prml 7299 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑔Q 𝑔 ∈ (1st𝐵))
8 rexex 2479 . . . . 5 (∃𝑔Q 𝑔 ∈ (1st𝐵) → ∃𝑔 𝑔 ∈ (1st𝐵))
96, 7, 83syl 17 . . . 4 (𝐵P → ∃𝑔 𝑔 ∈ (1st𝐵))
109ad2antlr 480 . . 3 (((𝐴P𝐵P) ∧ 𝑓 ∈ (1st𝐴)) → ∃𝑔 𝑔 ∈ (1st𝐵))
11 genpelvl.1 . . . . . . 7 𝐹 = (𝑤P, 𝑣P ↦ ⟨{𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (1st𝑤) ∧ 𝑧 ∈ (1st𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥Q ∣ ∃𝑦Q𝑧Q (𝑦 ∈ (2nd𝑤) ∧ 𝑧 ∈ (2nd𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
12 genpelvl.2 . . . . . . 7 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
1311, 12genpprecll 7336 . . . . . 6 ((𝐴P𝐵P) → ((𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵)) → (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵))))
1413imp 123 . . . . 5 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵)))
15 elprnql 7303 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
161, 15sylan 281 . . . . . . . . 9 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
17 elprnql 7303 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑔 ∈ (1st𝐵)) → 𝑔Q)
186, 17sylan 281 . . . . . . . . 9 ((𝐵P𝑔 ∈ (1st𝐵)) → 𝑔Q)
1916, 18anim12i 336 . . . . . . . 8 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (𝑓Q𝑔Q))
2019an4s 577 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓Q𝑔Q))
2112caovcl 5925 . . . . . . 7 ((𝑓Q𝑔Q) → (𝑓𝐺𝑔) ∈ Q)
2220, 21syl 14 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓𝐺𝑔) ∈ Q)
23 simpr 109 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → 𝑞 = (𝑓𝐺𝑔))
2423eleq1d 2208 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵))))
2522, 24rspcedv 2793 . . . . 5 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → ((𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵)) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵))))
2614, 25mpd 13 . . . 4 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
2726anassrs 397 . . 3 ((((𝐴P𝐵P) ∧ 𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st𝐵)) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
2810, 27exlimddv 1870 . 2 (((𝐴P𝐵P) ∧ 𝑓 ∈ (1st𝐴)) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
295, 28exlimddv 1870 1 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 962   = wceq 1331  wex 1468  wcel 1480  wrex 2417  {crab 2420  cop 3530  cfv 5123  (class class class)co 5774  cmpo 5776  1st c1st 6036  2nd c2nd 6037  Qcnq 7102  Pcnp 7113
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7126  df-nqqs 7170  df-inp 7288
This theorem is referenced by:  addclpr  7359  mulclpr  7394
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