Theorem genpnmax 11002

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 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
genpnmax.2	⊢ (𝑣 ∈ Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
genpnmax.3	⊢ (𝑧𝐺𝑤) = (𝑤𝐺𝑧)

Assertion

Ref	Expression
genpnmax	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (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 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		genp.1	. . 3 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ {𝑥 ∣ ∃𝑦 ∈ 𝑤 ∃𝑧 ∈ 𝑣 𝑥 = (𝑦𝐺𝑧)})
2		genp.2	. . 3 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
3	1, 2	genpelv 10995	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)))
4		prnmax 10990	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑔 <Q 𝑦)
5	4	adantr 482	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ∃𝑦 ∈ 𝐴 𝑔 <Q 𝑦)
6	1, 2	genpprecl 10996	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑦 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))
7	6	exp4b 432	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (𝑦 ∈ 𝐴 → (ℎ ∈ 𝐵 → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))))
8	7	com34 91	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (ℎ ∈ 𝐵 → (𝑦 ∈ 𝐴 → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))))
9	8	imp32 420	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑦 ∈ 𝐴 → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))
10		elprnq 10986	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
11		vex 3479	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
12		vex 3479	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
13		genpnmax.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
14		vex 3479	. . . . . . . . . . . . . . . 16 ⊢ ℎ ∈ V
15		genpnmax.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑧𝐺𝑤) = (𝑤𝐺𝑧)
16	11, 12, 13, 14, 15	caovord2 7619	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ Q → (𝑔 <Q 𝑦 ↔ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
17	16	biimpd 228	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ Q → (𝑔 <Q 𝑦 → (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
18	10, 17	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → (𝑔 <Q 𝑦 → (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
19	18	adantl 483	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑔 <Q 𝑦 → (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
20	9, 19	anim12d 610	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑦 ∈ 𝐴 ∧ 𝑔 <Q 𝑦) → ((𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ))))
21		breq2 5153	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦𝐺ℎ) → ((𝑔𝐺ℎ) <Q 𝑥 ↔ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
22	21	rspcev 3613	. . . . . . . . . . 11 ⊢ (((𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)
23	20, 22	syl6 35	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑦 ∈ 𝐴 ∧ 𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
24	23	adantlr 714	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑦 ∈ 𝐴 ∧ 𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
25	24	expd 417	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑦 ∈ 𝐴 → (𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)))
26	25	rexlimdv 3154	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (∃𝑦 ∈ 𝐴 𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
27	5, 26	mpd 15	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)
28	27	an4s 659	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)
29		breq1 5152	. . . . . 6 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺ℎ) <Q 𝑥))
30	29	rexbidv 3179	. . . . 5 ⊢ (𝑓 = (𝑔𝐺ℎ) → (∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥 ↔ ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
31	28, 30	imbitrrid 245	. . . 4 ⊢ (𝑓 = (𝑔𝐺ℎ) → (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
32	31	expdcom 416	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑓 = (𝑔𝐺ℎ) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)))
33	32	rexlimdvv 3211	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
34	3, 33	sylbid 239	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  ∃wrex 3071   class class class wbr 5149  (class class class)co 7409   ∈ cmpo 7411  Qcnq 10847   <_Q cltq 10853  Pcnp 10854

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 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636

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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-ni 10867  df-nq 10907  df-np 10976

This theorem is referenced by:  genpcl  11003