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Theorem genpnmax 9781
Description: An operation on positive reals has no largest member. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
genpnmax.2 (𝑣Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
genpnmax.3 (𝑧𝐺𝑤) = (𝑤𝐺𝑧)
Assertion
Ref Expression
genpnmax ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑓   𝑓,𝐹,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐹(𝑧,𝑤,𝑣)

Proof of Theorem genpnmax
Dummy variables 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 genp.1 . . 3 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
2 genp.2 . . 3 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31, 2genpelv 9774 . 2 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
4 prnmax 9769 . . . . . . . 8 ((𝐴P𝑔𝐴) → ∃𝑦𝐴 𝑔 <Q 𝑦)
54adantr 481 . . . . . . 7 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ∃𝑦𝐴 𝑔 <Q 𝑦)
61, 2genpprecl 9775 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P) → ((𝑦𝐴𝐵) → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))
76exp4b 631 . . . . . . . . . . . . . 14 (𝐴P → (𝐵P → (𝑦𝐴 → (𝐵 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))))
87com34 91 . . . . . . . . . . . . 13 (𝐴P → (𝐵P → (𝐵 → (𝑦𝐴 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))))
98imp32 449 . . . . . . . . . . . 12 ((𝐴P ∧ (𝐵P𝐵)) → (𝑦𝐴 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))
10 elprnq 9765 . . . . . . . . . . . . . 14 ((𝐵P𝐵) → Q)
11 vex 3192 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
12 vex 3192 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
13 genpnmax.2 . . . . . . . . . . . . . . . 16 (𝑣Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
14 vex 3192 . . . . . . . . . . . . . . . 16 ∈ V
15 genpnmax.3 . . . . . . . . . . . . . . . 16 (𝑧𝐺𝑤) = (𝑤𝐺𝑧)
1611, 12, 13, 14, 15caovord2 6806 . . . . . . . . . . . . . . 15 (Q → (𝑔 <Q 𝑦 ↔ (𝑔𝐺) <Q (𝑦𝐺)))
1716biimpd 219 . . . . . . . . . . . . . 14 (Q → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
1810, 17syl 17 . . . . . . . . . . . . 13 ((𝐵P𝐵) → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
1918adantl 482 . . . . . . . . . . . 12 ((𝐴P ∧ (𝐵P𝐵)) → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
209, 19anim12d 585 . . . . . . . . . . 11 ((𝐴P ∧ (𝐵P𝐵)) → ((𝑦𝐴𝑔 <Q 𝑦) → ((𝑦𝐺) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺) <Q (𝑦𝐺))))
21 breq2 4622 . . . . . . . . . . . 12 (𝑥 = (𝑦𝐺) → ((𝑔𝐺) <Q 𝑥 ↔ (𝑔𝐺) <Q (𝑦𝐺)))
2221rspcev 3298 . . . . . . . . . . 11 (((𝑦𝐺) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺) <Q (𝑦𝐺)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)
2320, 22syl6 35 . . . . . . . . . 10 ((𝐴P ∧ (𝐵P𝐵)) → ((𝑦𝐴𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
2423adantlr 750 . . . . . . . . 9 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑦𝐴𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
2524expd 452 . . . . . . . 8 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑦𝐴 → (𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)))
2625rexlimdv 3024 . . . . . . 7 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (∃𝑦𝐴 𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
275, 26mpd 15 . . . . . 6 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)
2827an4s 868 . . . . 5 (((𝐴P𝐵P) ∧ (𝑔𝐴𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)
29 breq1 4621 . . . . . 6 (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺) <Q 𝑥))
3029rexbidv 3046 . . . . 5 (𝑓 = (𝑔𝐺) → (∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥 ↔ ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
3128, 30syl5ibr 236 . . . 4 (𝑓 = (𝑔𝐺) → (((𝐴P𝐵P) ∧ (𝑔𝐴𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
3231expdcom 455 . . 3 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑓 = (𝑔𝐺) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)))
3332rexlimdvv 3031 . 2 ((𝐴P𝐵P) → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
343, 33sylbid 230 1 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wrex 2908   class class class wbr 4618  (class class class)co 6610  cmpt2 6612  Qcnq 9626   <Q cltq 9632  Pcnp 9633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-inf2 8490
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-ni 9646  df-nq 9686  df-np 9755
This theorem is referenced by:  genpcl  9782
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