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Theorem genpnmax 10013
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 10006 . 2 ((𝐴P𝐵P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)))
4 prnmax 10001 . . . . . . . 8 ((𝐴P𝑔𝐴) → ∃𝑦𝐴 𝑔 <Q 𝑦)
54adantr 472 . . . . . . 7 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ∃𝑦𝐴 𝑔 <Q 𝑦)
61, 2genpprecl 10007 . . . . . . . . . . . . . . 15 ((𝐴P𝐵P) → ((𝑦𝐴𝐵) → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))
76exp4b 633 . . . . . . . . . . . . . 14 (𝐴P → (𝐵P → (𝑦𝐴 → (𝐵 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))))
87com34 91 . . . . . . . . . . . . 13 (𝐴P → (𝐵P → (𝐵 → (𝑦𝐴 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))))
98imp32 448 . . . . . . . . . . . 12 ((𝐴P ∧ (𝐵P𝐵)) → (𝑦𝐴 → (𝑦𝐺) ∈ (𝐴𝐹𝐵)))
10 elprnq 9997 . . . . . . . . . . . . . 14 ((𝐵P𝐵) → Q)
11 vex 3335 . . . . . . . . . . . . . . . 16 𝑔 ∈ V
12 vex 3335 . . . . . . . . . . . . . . . 16 𝑦 ∈ V
13 genpnmax.2 . . . . . . . . . . . . . . . 16 (𝑣Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
14 vex 3335 . . . . . . . . . . . . . . . 16 ∈ V
15 genpnmax.3 . . . . . . . . . . . . . . . 16 (𝑧𝐺𝑤) = (𝑤𝐺𝑧)
1611, 12, 13, 14, 15caovord2 7003 . . . . . . . . . . . . . . 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 473 . . . . . . . . . . . 12 ((𝐴P ∧ (𝐵P𝐵)) → (𝑔 <Q 𝑦 → (𝑔𝐺) <Q (𝑦𝐺)))
209, 19anim12d 587 . . . . . . . . . . 11 ((𝐴P ∧ (𝐵P𝐵)) → ((𝑦𝐴𝑔 <Q 𝑦) → ((𝑦𝐺) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺) <Q (𝑦𝐺))))
21 breq2 4800 . . . . . . . . . . . 12 (𝑥 = (𝑦𝐺) → ((𝑔𝐺) <Q 𝑥 ↔ (𝑔𝐺) <Q (𝑦𝐺)))
2221rspcev 3441 . . . . . . . . . . 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 753 . . . . . . . . 9 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ((𝑦𝐴𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
2524expd 451 . . . . . . . 8 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (𝑦𝐴 → (𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)))
2625rexlimdv 3160 . . . . . . 7 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → (∃𝑦𝐴 𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥))
275, 26mpd 15 . . . . . 6 (((𝐴P𝑔𝐴) ∧ (𝐵P𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)
2827an4s 904 . . . . 5 (((𝐴P𝐵P) ∧ (𝑔𝐴𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺) <Q 𝑥)
29 breq1 4799 . . . . . 6 (𝑓 = (𝑔𝐺) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺) <Q 𝑥))
3029rexbidv 3182 . . . . 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 454 . . 3 ((𝐴P𝐵P) → ((𝑔𝐴𝐵) → (𝑓 = (𝑔𝐺) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)))
3332rexlimdvv 3167 . 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 383   = wceq 1624  wcel 2131  {cab 2738  wrex 3043   class class class wbr 4796  (class class class)co 6805  cmpt2 6807  Qcnq 9858   <Q cltq 9864  Pcnp 9865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-inf2 8703
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-om 7223  df-ni 9878  df-nq 9918  df-np 9987
This theorem is referenced by:  genpcl  10014
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