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Theorem genpnmax 10950
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 10943 . 2 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
4 prnmax 10938 . . . . . . . 8 ((𝐴P𝑔𝐴) → ∃𝑦𝐴 𝑔 <Q 𝑦)
54adantr 482 . . . . . . 7 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ∃𝑦𝐴 𝑔 <Q 𝑦)
61, 2genpprecl 10944 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P) → ((𝑦𝐴𝐵) → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))
76exp4b 432 . . . . . . . . . . . . . 14 (𝐴P → (𝐵P → (𝑦𝐴 → (𝐵 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))))
87com34 91 . . . . . . . . . . . . 13 (𝐴P → (𝐵P → (𝐵 → (𝑦𝐴 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))))
98imp32 420 . . . . . . . . . . . 12 ((𝐴P ∧ (𝐵P𝐵)) → (𝑦𝐴 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))
10 elprnq 10934 . . . . . . . . . . . . . 14 ((𝐵P𝐵) → Q)
11 vex 3452 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
12 vex 3452 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
13 genpnmax.2 . . . . . . . . . . . . . . . 16 (𝑣Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
14 vex 3452 . . . . . . . . . . . . . . . 16 ∈ V
15 genpnmax.3 . . . . . . . . . . . . . . . 16 (𝑧𝐺𝑤) = (𝑤𝐺𝑧)
1611, 12, 13, 14, 15caovord2 7571 . . . . . . . . . . . . . . 15 (Q → (𝑔 <Q 𝑦 ↔ (𝑔𝐺) <Q (𝑦𝐺)))
1716biimpd 228 . . . . . . . . . . . . . 14 (Q → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
1810, 17syl 17 . . . . . . . . . . . . 13 ((𝐵P𝐵) → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
1918adantl 483 . . . . . . . . . . . 12 ((𝐴P ∧ (𝐵P𝐵)) → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
209, 19anim12d 610 . . . . . . . . . . 11 ((𝐴P ∧ (𝐵P𝐵)) → ((𝑦𝐴𝑔 <Q 𝑦) → ((𝑦𝐺) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺) <Q (𝑦𝐺))))
21 breq2 5114 . . . . . . . . . . . 12 (𝑥 = (𝑦𝐺) → ((𝑔𝐺) <Q 𝑥 ↔ (𝑔𝐺) <Q (𝑦𝐺)))
2221rspcev 3584 . . . . . . . . . . 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 714 . . . . . . . . 9 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑦𝐴𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
2524expd 417 . . . . . . . 8 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑦𝐴 → (𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)))
2625rexlimdv 3151 . . . . . . 7 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (∃𝑦𝐴 𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
275, 26mpd 15 . . . . . 6 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)
2827an4s 659 . . . . 5 (((𝐴P𝐵P) ∧ (𝑔𝐴𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)
29 breq1 5113 . . . . . 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, 30syl5ibr 246 . . . 4 (𝑓 = (𝑔𝐺) → (((𝐴P𝐵P) ∧ (𝑔𝐴𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
3231expdcom 416 . . 3 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑓 = (𝑔𝐺) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)))
3332rexlimdvv 3205 . 2 ((𝐴P𝐵P) → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
343, 33sylbid 239 1 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {cab 2714  wrex 3074   class class class wbr 5110  (class class class)co 7362  cmpo 7364  Qcnq 10795   <Q cltq 10801  Pcnp 10802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-ni 10815  df-nq 10855  df-np 10924
This theorem is referenced by:  genpcl  10951
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