Theorem genpnmax 10918

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 10911	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔 ∈ 𝐴 ∃ℎ ∈ 𝐵 𝑓 = (𝑔𝐺ℎ)))
4		prnmax 10906	. . . . . . . 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 10912	. . . . . . . . . . . . . . 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 10902	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ 𝐵) → ℎ ∈ Q)
11		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
12		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
13		genpnmax.2	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ Q → (𝑧 <Q 𝑤 ↔ (𝑣𝐺𝑧) <Q (𝑣𝐺𝑤)))
14		vex 3444	. . . . . . . . . . . . . . . 16 ⊢ ℎ ∈ V
15		genpnmax.3	. . . . . . . . . . . . . . . 16 ⊢ (𝑧𝐺𝑤) = (𝑤𝐺𝑧)
16	11, 12, 13, 14, 15	caovord2 7570	. . . . . . . . . . . . . . 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 5102	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑦𝐺ℎ) → ((𝑔𝐺ℎ) <Q 𝑥 ↔ (𝑔𝐺ℎ) <Q (𝑦𝐺ℎ)))
22	21	rspcev 3576	. . . . . . . . . . 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 3135	. . . . . . 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 5101	. . . . . 6 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺ℎ) <Q 𝑥))
30	29	rexbidv 3160	. . . . 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 3192	. 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 2113  {cab 2714  ∃wrex 3060   class class class wbr 5098  (class class class)co 7358   ∈ cmpo 7360  Qcnq 10763   <_Q cltq 10769  Pcnp 10770

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-inf2 9550

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-ni 10783  df-nq 10823  df-np 10892

This theorem is referenced by:  genpcl  10919