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Theorem genpml 7466
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 7424 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prml 7426 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑓Q 𝑓 ∈ (1st𝐴))
3 rexex 2516 . . . 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 7424 . . . . 5 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
7 prml 7426 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑔Q 𝑔 ∈ (1st𝐵))
8 rexex 2516 . . . . 5 (∃𝑔Q 𝑔 ∈ (1st𝐵) → ∃𝑔 𝑔 ∈ (1st𝐵))
96, 7, 83syl 17 . . . 4 (𝐵P → ∃𝑔 𝑔 ∈ (1st𝐵))
109ad2antlr 486 . . 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 7463 . . . . . 6 ((𝐴P𝐵P) → ((𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵)) → (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵))))
1413imp 123 . . . . 5 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵)))
15 elprnql 7430 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
161, 15sylan 281 . . . . . . . . 9 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
17 elprnql 7430 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑔 ∈ (1st𝐵)) → 𝑔Q)
186, 17sylan 281 . . . . . . . . 9 ((𝐵P𝑔 ∈ (1st𝐵)) → 𝑔Q)
1916, 18anim12i 336 . . . . . . . 8 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (𝑓Q𝑔Q))
2019an4s 583 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓Q𝑔Q))
2112caovcl 6004 . . . . . . 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 2239 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵))))
2522, 24rspcedv 2838 . . . . 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 398 . . 3 ((((𝐴P𝐵P) ∧ 𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st𝐵)) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
2810, 27exlimddv 1891 . 2 (((𝐴P𝐵P) ∧ 𝑓 ∈ (1st𝐴)) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
295, 28exlimddv 1891 1 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 973   = wceq 1348  wex 1485  wcel 2141  wrex 2449  {crab 2452  cop 3584  cfv 5196  (class class class)co 5850  cmpo 5852  1st c1st 6114  2nd c2nd 6115  Qcnq 7229  Pcnp 7240
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-ov 5853  df-oprab 5854  df-mpo 5855  df-1st 6116  df-2nd 6117  df-qs 6515  df-ni 7253  df-nqqs 7297  df-inp 7415
This theorem is referenced by:  addclpr  7486  mulclpr  7521
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