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Theorem genpnmax 11044
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 11037 . 2 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
4 prnmax 11032 . . . . . . . 8 ((𝐴P𝑔𝐴) → ∃𝑦𝐴 𝑔 <Q 𝑦)
54adantr 480 . . . . . . 7 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ∃𝑦𝐴 𝑔 <Q 𝑦)
61, 2genpprecl 11038 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P) → ((𝑦𝐴𝐵) → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))
76exp4b 430 . . . . . . . . . . . . . 14 (𝐴P → (𝐵P → (𝑦𝐴 → (𝐵 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))))
87com34 91 . . . . . . . . . . . . 13 (𝐴P → (𝐵P → (𝐵 → (𝑦𝐴 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))))
98imp32 418 . . . . . . . . . . . 12 ((𝐴P ∧ (𝐵P𝐵)) → (𝑦𝐴 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))
10 elprnq 11028 . . . . . . . . . . . . . 14 ((𝐵P𝐵) → Q)
11 vex 3481 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
12 vex 3481 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
13 genpnmax.2 . . . . . . . . . . . . . . . 16 (𝑣Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
14 vex 3481 . . . . . . . . . . . . . . . 16 ∈ V
15 genpnmax.3 . . . . . . . . . . . . . . . 16 (𝑧𝐺𝑤) = (𝑤𝐺𝑧)
1611, 12, 13, 14, 15caovord2 7644 . . . . . . . . . . . . . . 15 (Q → (𝑔 <Q 𝑦 ↔ (𝑔𝐺) <Q (𝑦𝐺)))
1716biimpd 229 . . . . . . . . . . . . . 14 (Q → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
1810, 17syl 17 . . . . . . . . . . . . 13 ((𝐵P𝐵) → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
1918adantl 481 . . . . . . . . . . . 12 ((𝐴P ∧ (𝐵P𝐵)) → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
209, 19anim12d 609 . . . . . . . . . . 11 ((𝐴P ∧ (𝐵P𝐵)) → ((𝑦𝐴𝑔 <Q 𝑦) → ((𝑦𝐺) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺) <Q (𝑦𝐺))))
21 breq2 5151 . . . . . . . . . . . 12 (𝑥 = (𝑦𝐺) → ((𝑔𝐺) <Q 𝑥 ↔ (𝑔𝐺) <Q (𝑦𝐺)))
2221rspcev 3621 . . . . . . . . . . 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 715 . . . . . . . . 9 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑦𝐴𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
2524expd 415 . . . . . . . 8 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑦𝐴 → (𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)))
2625rexlimdv 3150 . . . . . . 7 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (∃𝑦𝐴 𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
275, 26mpd 15 . . . . . 6 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)
2827an4s 660 . . . . 5 (((𝐴P𝐵P) ∧ (𝑔𝐴𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)
29 breq1 5150 . . . . . 6 (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺) <Q 𝑥))
3029rexbidv 3176 . . . . 5 (𝑓 = (𝑔𝐺) → (∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥 ↔ ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
3128, 30imbitrrid 246 . . . 4 (𝑓 = (𝑔𝐺) → (((𝐴P𝐵P) ∧ (𝑔𝐴𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
3231expdcom 414 . . 3 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑓 = (𝑔𝐺) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)))
3332rexlimdvv 3209 . 2 ((𝐴P𝐵P) → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
343, 33sylbid 240 1 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  {cab 2711  wrex 3067   class class class wbr 5147  (class class class)co 7430  cmpo 7432  Qcnq 10889   <Q cltq 10895  Pcnp 10896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-ni 10909  df-nq 10949  df-np 11018
This theorem is referenced by:  genpcl  11045
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