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Theorem genpml 6613
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 6571 . . . 4 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prml 6573 . . . 4 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑓Q 𝑓 ∈ (1st𝐴))
3 rexex 2368 . . . 4 (∃𝑓Q 𝑓 ∈ (1st𝐴) → ∃𝑓 𝑓 ∈ (1st𝐴))
41, 2, 33syl 17 . . 3 (𝐴P → ∃𝑓 𝑓 ∈ (1st𝐴))
54adantr 261 . 2 ((𝐴P𝐵P) → ∃𝑓 𝑓 ∈ (1st𝐴))
6 prop 6571 . . . . 5 (𝐵P → ⟨(1st𝐵), (2nd𝐵)⟩ ∈ P)
7 prml 6573 . . . . 5 (⟨(1st𝐵), (2nd𝐵)⟩ ∈ P → ∃𝑔Q 𝑔 ∈ (1st𝐵))
8 rexex 2368 . . . . 5 (∃𝑔Q 𝑔 ∈ (1st𝐵) → ∃𝑔 𝑔 ∈ (1st𝐵))
96, 7, 83syl 17 . . . 4 (𝐵P → ∃𝑔 𝑔 ∈ (1st𝐵))
109ad2antlr 458 . . 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 6610 . . . . . 6 ((𝐴P𝐵P) → ((𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵)) → (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵))))
1413imp 115 . . . . 5 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵)))
15 elprnql 6577 . . . . . . . . . 10 ((⟨(1st𝐴), (2nd𝐴)⟩ ∈ P𝑓 ∈ (1st𝐴)) → 𝑓Q)
161, 15sylan 267 . . . . . . . . 9 ((𝐴P𝑓 ∈ (1st𝐴)) → 𝑓Q)
17 elprnql 6577 . . . . . . . . . 10 ((⟨(1st𝐵), (2nd𝐵)⟩ ∈ P𝑔 ∈ (1st𝐵)) → 𝑔Q)
186, 17sylan 267 . . . . . . . . 9 ((𝐵P𝑔 ∈ (1st𝐵)) → 𝑔Q)
1916, 18anim12i 321 . . . . . . . 8 (((𝐴P𝑓 ∈ (1st𝐴)) ∧ (𝐵P𝑔 ∈ (1st𝐵))) → (𝑓Q𝑔Q))
2019an4s 522 . . . . . . 7 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓Q𝑔Q))
2112caovcl 5655 . . . . . . 7 ((𝑓Q𝑔Q) → (𝑓𝐺𝑔) ∈ Q)
2220, 21syl 14 . . . . . 6 (((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) → (𝑓𝐺𝑔) ∈ Q)
23 simpr 103 . . . . . . 7 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → 𝑞 = (𝑓𝐺𝑔))
2423eleq1d 2106 . . . . . 6 ((((𝐴P𝐵P) ∧ (𝑓 ∈ (1st𝐴) ∧ 𝑔 ∈ (1st𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (1st ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (1st ‘(𝐴𝐹𝐵))))
2522, 24rspcedv 2660 . . . . 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 380 . . 3 ((((𝐴P𝐵P) ∧ 𝑓 ∈ (1st𝐴)) ∧ 𝑔 ∈ (1st𝐵)) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
2810, 27exlimddv 1778 . 2 (((𝐴P𝐵P) ∧ 𝑓 ∈ (1st𝐴)) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
295, 28exlimddv 1778 1 ((𝐴P𝐵P) → ∃𝑞Q 𝑞 ∈ (1st ‘(𝐴𝐹𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  w3a 885   = wceq 1243  wex 1381  wcel 1393  wrex 2307  {crab 2310  cop 3378  cfv 4902  (class class class)co 5512  cmpt2 5514  1st c1st 5765  2nd c2nd 5766  Qcnq 6376  Pcnp 6387
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-qs 6112  df-ni 6400  df-nqqs 6444  df-inp 6562
This theorem is referenced by:  addclpr  6633  mulclpr  6668
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