Theorem genpnmax 11119

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 11112	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)))
4		prnmax 11107	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) → ∃𝑦 ∈ 𝐴 𝑔 <Q 𝑦)
5	4	adantr 486	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ∃𝑦 ∈ 𝐴 𝑔 <Q 𝑦)
6	1, 2	genpprecl 11113	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑦 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))
7	6	exp4b 436	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (𝑦 ∈ 𝐴 → (ℎ ∈ 𝐵 → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))))
8	7	com34 92	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ P → (𝐵 ∈ P → (ℎ ∈ 𝐵 → (𝑦 ∈ 𝐴 → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))))
9	8	imp32 424	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑦 ∈ 𝐴 → (𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵)))
10		elprnq 11103	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
11		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
12		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
13		genpnmax.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
14		vex 3458	. . . . . . . . . . . . . . . 16 ⊢ ℎ ∈ V
15		genpnmax.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑧𝐺𝑤) = (𝑤𝐺𝑧)
16	11, 12, 13, 14, 15	caovord2 7657	. . . . . . . . . . . . . . 15 ⊢ (ℎ ∈ Q → (𝑔 <Q 𝑦 ↔ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
17	16	biimpd 232	. . . . . . . . . . . . . 14 ⊢ (ℎ ∈ Q → (𝑔 <Q 𝑦 → (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
18	10, 17	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → (𝑔 <Q 𝑦 → (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
19	18	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑔 <Q 𝑦 → (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
20	9, 19	anim12d 621	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑦 ∈ 𝐴 ∧ 𝑔 <Q 𝑦) → ((𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ))))
21		breq2 5110	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦𝐺ℎ) → ((𝑔𝐺ℎ) <Q 𝑥 ↔ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
22	21	rspcev 3580	. . . . . . . . . . 11 ⊢ (((𝑦𝐺ℎ) ∈ (𝐴𝐹𝐵) ∧ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)
23	20, 22	syl6 36	. . . . . . . . . 10 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑦 ∈ 𝐴 ∧ 𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
24	23	adantlr 728	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ((𝑦 ∈ 𝐴 ∧ 𝑔 <Q 𝑦) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
25	24	expd 421	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (𝑦 ∈ 𝐴 → (𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)))
26	25	rexlimdv 3165	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → (∃𝑦 ∈ 𝐴 𝑔 <Q 𝑦 → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
27	5, 26	mpd 16	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ 𝐴) ∧ (𝐵 ∈ P ∧ ℎ ∈ 𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)
28	27	an4s 673	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥)
29		breq1 5109	. . . . . 6 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺ℎ) <Q 𝑥))
30	29	rexbidv 3190	. . . . 5 ⊢ (𝑓 = (𝑔𝐺ℎ) → (∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥 ↔ ∃𝑥 ∈ (𝐴𝐹𝐵)(𝑔𝐺ℎ) <Q 𝑥))
31	28, 30	imbitrrid 249	. . . 4 ⊢ (𝑓 = (𝑔𝐺ℎ) → (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵)) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
32	31	expdcom 420	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑔 ∈ 𝐴 ∧ ℎ ∈ 𝐵) → (𝑓 = (𝑔𝐺ℎ) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥)))
33	32	rexlimdvv 3222	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))
34	3, 33	sylbid 243	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) → ∃𝑥 ∈ (𝐴𝐹𝐵)𝑓 <Q 𝑥))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090   class class class wbr 5106  (class class class)co 7443   ∈ cmpo 7445  Qcnq 10964   <_Q cltq 10970  Pcnp 10971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-ni 10984  df-nq 11024  df-np 11093

This theorem is used by:  genpcl  11120