Theorem genpcuu 6618

Description: Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)

Hypotheses

Ref	Expression
genpelvl.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
genpcuu.2	⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))

Assertion

Ref	Expression
genpcuu	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑓,𝑔,ℎ,𝑤,𝑣,𝐴   𝑥,𝐵,𝑦,𝑧,𝑓,𝑔,ℎ,𝑤,𝑣   𝑥,𝐺,𝑦,𝑧,𝑓,𝑔,ℎ,𝑤,𝑣   𝑓,𝐹,𝑔,ℎ

Allowed substitution hints:   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem genpcuu

Step	Hyp	Ref	Expression
1		ltrelnq 6463	. . . . . . 7 ⊢ <Q ⊆ (Q × Q)
2	1	brel 4392	. . . . . 6 ⊢ (𝑓 <Q 𝑥 → (𝑓 ∈ Q ∧ 𝑥 ∈ Q))
3	2	simprd 107	. . . . 5 ⊢ (𝑓 <Q 𝑥 → 𝑥 ∈ Q)
4		genpelvl.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
5		genpelvl.2	. . . . . . . . 9 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
6	4, 5	genpelvu 6611	. . . . . . . 8 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
7	6	adantr 261	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)))
8		breq1 3767	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 ↔ (𝑔𝐺ℎ) <Q 𝑥))
9	8	biimpd 132	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 → (𝑔𝐺ℎ) <Q 𝑥))
10		genpcuu.2	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) → ((𝑔𝐺ℎ) <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))
11	9, 10	sylan9r 390	. . . . . . . . . . 11 ⊢ (((((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) ∧ 𝑥 ∈ Q) ∧ 𝑓 = (𝑔𝐺ℎ)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))
12	11	exp31 346	. . . . . . . . . 10 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (2nd ‘𝐵))) → (𝑥 ∈ Q → (𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
13	12	an4s 522	. . . . . . . . 9 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ (2nd ‘𝐴) ∧ ℎ ∈ (2nd ‘𝐵))) → (𝑥 ∈ Q → (𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
14	13	impancom 247	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → ((𝑔 ∈ (2nd ‘𝐴) ∧ ℎ ∈ (2nd ‘𝐵)) → (𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
15	14	rexlimdvv 2439	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (∃𝑔 ∈ (2nd ‘𝐴)∃ℎ ∈ (2nd ‘𝐵)𝑓 = (𝑔𝐺ℎ) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
16	7, 15	sylbid 139	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑥 ∈ Q) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
17	16	ex 108	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑥 ∈ Q → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
18	3, 17	syl5 28	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 <Q 𝑥 → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
19	18	com34 77	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 <Q 𝑥 → (𝑓 <Q 𝑥 → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵))))))
20	19	pm2.43d 44	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 <Q 𝑥 → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))
21	20	com23 72	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝑓 ∈ (2nd ‘(𝐴𝐹𝐵)) → (𝑓 <Q 𝑥 → 𝑥 ∈ (2nd ‘(𝐴𝐹𝐵)))))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∃wrex 2307  {crab 2310  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  (class class class)co 5512   ↦ cmpt2 5514  1^st c1st 5765  2^nd c2nd 5766  Qcnq 6378   <_Q cltq 6383  Pcnp 6389

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-qs 6112  df-ni 6402  df-nqqs 6446  df-ltnqqs 6451  df-inp 6564

This theorem is referenced by:  genprndu  6620