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Theorem genpml 7226
 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 7184 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prml 7186 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑓Q 𝑓 ∈ (1st𝐴))
3 rexex 2438 . . . 4 (∃𝑓Q 𝑓 ∈ (1st𝐴) → ∃𝑓 𝑓 ∈ (1st𝐴))
41, 2, 33syl 17 . . 3 (𝐴P → ∃𝑓 𝑓 ∈ (1st𝐴))
54adantr 272 . 2 ((𝐴P𝐵P) → ∃𝑓 𝑓 ∈ (1st𝐴))
6 prop 7184 . . . . 5 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
7 prml 7186 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑔Q 𝑔 ∈ (1st𝐵))
8 rexex 2438 . . . . 5 (∃𝑔Q 𝑔 ∈ (1st𝐵) → ∃𝑔 𝑔 ∈ (1st𝐵))
96, 7, 83syl 17 . . . 4 (𝐵P → ∃𝑔 𝑔 ∈ (1st𝐵))
109ad2antlr 476 . . 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 7223 . . . . . 6 ((𝐴P𝐵P) → ((𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵)) → (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵))))
1413imp 123 . . . . 5 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵)))
15 elprnql 7190 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
161, 15sylan 279 . . . . . . . . 9 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
17 elprnql 7190 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑔 ∈ (1st𝐵)) → 𝑔Q)
186, 17sylan 279 . . . . . . . . 9 ((𝐵P𝑔 ∈ (1st𝐵)) → 𝑔Q)
1916, 18anim12i 334 . . . . . . . 8 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (𝑓Q𝑔Q))
2019an4s 558 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓Q𝑔Q))
2112caovcl 5857 . . . . . . 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 2168 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵))))
2522, 24rspcedv 2748 . . . . 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 395 . . 3 ((((𝐴P𝐵P) ∧ 𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st𝐵)) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
2810, 27exlimddv 1837 . 2 (((𝐴P𝐵P) ∧ 𝑓 ∈ (1st𝐴)) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
295, 28exlimddv 1837 1 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 930   = wceq 1299  ∃wex 1436   ∈ wcel 1448  ∃wrex 2376  {crab 2379  ⟨cop 3477  ‘cfv 5059  (class class class)co 5706   ∈ cmpo 5708  1st c1st 5967  2nd c2nd 5968  Qcnq 6989  Pcnp 7000 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440 This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-qs 6365  df-ni 7013  df-nqqs 7057  df-inp 7175 This theorem is referenced by:  addclpr  7246  mulclpr  7281
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