Theorem genpnmax 10895

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 10888	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)))
4		prnmax 10883	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑔 <Q 𝑦)
5	4	adantr 480	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ∃𝑦 ∈ 𝐴 𝑔 <Q 𝑦)
6	1, 2	genpprecl 10889	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑦 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))
7	6	exp4b 430	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑦 ∈ 𝐴 → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))
10		elprnq 10879	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
11		vex 3440	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
12		vex 3440	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
13		genpnmax.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
14		vex 3440	. . . . . . . . . . . . . . . 16 ⊢ ℎ ∈ V
15		genpnmax.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑧𝐺𝑤) = (𝑤𝐺𝑧)
16	11, 12, 13, 14, 15	caovord2 7558	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ Q → (𝑔 <Q 𝑦 ↔ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
17	16	biimpd 229	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑔 <Q 𝑦 → (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
20	9, 19	anim12d 609	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑦 ∈ 𝐴 ∧ 𝑔 <Q 𝑦) → ((𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ))))
21		breq2 5095	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦𝐺ℎ) → ((𝑔𝐺ℎ) <Q 𝑥 ↔ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
22	21	rspcev 3577	. . . . . . . . . . 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 715	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑦 ∈ 𝐴 ∧ 𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
25	24	expd 415	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑦 ∈ 𝐴 → (𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)))
26	25	rexlimdv 3131	. . . . . . 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 660	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)
29		breq1 5094	. . . . . 6 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺ℎ) <Q 𝑥))
30	29	rexbidv 3156	. . . . 5 ⊢ (𝑓 = (𝑔𝐺ℎ) → (∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥 ↔ ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
31	28, 30	imbitrrid 246	. . . 4 ⊢ (𝑓 = (𝑔𝐺ℎ) → (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
32	31	expdcom 414	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑓 = (𝑔𝐺ℎ) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)))
33	32	rexlimdvv 3188	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
34	3, 33	sylbid 240	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∃wrex 3056   class class class wbr 5091  (class class class)co 7346   ∈ cmpo 7348  Qcnq 10740   <_Q cltq 10746  Pcnp 10747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-ni 10760  df-nq 10800  df-np 10869

This theorem is referenced by:  genpcl  10896