Theorem genpmu 6500

Description: The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-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)

Assertion

Ref	Expression
genpmu	⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B)))

Distinct variable groups:   x,y,z,w,v,𝑞,A   x,B,y,z,w,v,𝑞   x,𝐺,y,z,w,v,𝑞   𝐹,𝑞

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

Proof of Theorem genpmu

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 6457	. . . 4 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
2		prmu 6460	. . . 4 ⊢ (⟨(1st ‘A), (2nd ‘A)⟩ ∈ P → ∃f ∈ Q f ∈ (2nd ‘A))
3		rexex 2362	. . . 4 ⊢ (∃f ∈ Q f ∈ (2nd ‘A) → ∃f f ∈ (2nd ‘A))
4	1, 2, 3	3syl 17	. . 3 ⊢ (A ∈ P → ∃f f ∈ (2nd ‘A))
5	4	adantr 261	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → ∃f f ∈ (2nd ‘A))
6		prop 6457	. . . . 5 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
7		prmu 6460	. . . . 5 ⊢ (⟨(1st ‘B), (2nd ‘B)⟩ ∈ P → ∃g ∈ Q g ∈ (2nd ‘B))
8		rexex 2362	. . . . 5 ⊢ (∃g ∈ Q g ∈ (2nd ‘B) → ∃g g ∈ (2nd ‘B))
9	6, 7, 8	3syl 17	. . . 4 ⊢ (B ∈ P → ∃g g ∈ (2nd ‘B))
10	9	ad2antlr 458	. . 3 ⊢ (((A ∈ P ∧ B ∈ P) ∧ f ∈ (2nd ‘A)) → ∃g g ∈ (2nd ‘B))
11		genpelvl.1	. . . . . . 7 ⊢ 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))}⟩)
12		genpelvl.2	. . . . . . 7 ⊢ ((y ∈ Q ∧ z ∈ Q) → (y𝐺z) ∈ Q)
13	11, 12	genppreclu 6497	. . . . . 6 ⊢ ((A ∈ P ∧ B ∈ P) → ((f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B)) → (f𝐺g) ∈ (2nd ‘(A𝐹B))))
14	13	imp 115	. . . . 5 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B))) → (f𝐺g) ∈ (2nd ‘(A𝐹B)))
15		elprnqu 6464	. . . . . . . . . 10 ⊢ ((⟨(1st ‘A), (2nd ‘A)⟩ ∈ P ∧ f ∈ (2nd ‘A)) → f ∈ Q)
16	1, 15	sylan 267	. . . . . . . . 9 ⊢ ((A ∈ P ∧ f ∈ (2nd ‘A)) → f ∈ Q)
17		elprnqu 6464	. . . . . . . . . 10 ⊢ ((⟨(1st ‘B), (2nd ‘B)⟩ ∈ P ∧ g ∈ (2nd ‘B)) → g ∈ Q)
18	6, 17	sylan 267	. . . . . . . . 9 ⊢ ((B ∈ P ∧ g ∈ (2nd ‘B)) → g ∈ Q)
19	16, 18	anim12i 321	. . . . . . . 8 ⊢ (((A ∈ P ∧ f ∈ (2nd ‘A)) ∧ (B ∈ P ∧ g ∈ (2nd ‘B))) → (f ∈ Q ∧ g ∈ Q))
20	19	an4s 522	. . . . . . 7 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B))) → (f ∈ Q ∧ g ∈ Q))
21	12	caovcl 5597	. . . . . . 7 ⊢ ((f ∈ Q ∧ g ∈ Q) → (f𝐺g) ∈ Q)
22	20, 21	syl 14	. . . . . 6 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B))) → (f𝐺g) ∈ Q)
23		simpr 103	. . . . . . 7 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B))) ∧ 𝑞 = (f𝐺g)) → 𝑞 = (f𝐺g))
24	23	eleq1d 2103	. . . . . 6 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B))) ∧ 𝑞 = (f𝐺g)) → (𝑞 ∈ (2nd ‘(A𝐹B)) ↔ (f𝐺g) ∈ (2nd ‘(A𝐹B))))
25	22, 24	rspcedv 2654	. . . . 5 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B))) → ((f𝐺g) ∈ (2nd ‘(A𝐹B)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B))))
26	14, 25	mpd 13	. . . 4 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B))) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B)))
27	26	anassrs 380	. . 3 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ f ∈ (2nd ‘A)) ∧ g ∈ (2nd ‘B)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B)))
28	10, 27	exlimddv 1775	. 2 ⊢ (((A ∈ P ∧ B ∈ P) ∧ f ∈ (2nd ‘A)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B)))
29	5, 28	exlimddv 1775	1 ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B)))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  {crab 2304  ⟨cop 3370  ‘cfv 4845  (class class class)co 5455   ↦ cmpt2 5457  1^st c1st 5707  2^nd c2nd 5708  Qcnq 6264  Pcnp 6275

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-qs 6048  df-ni 6288  df-nqqs 6332  df-inp 6448

This theorem is referenced by:  addclpr  6519  mulclpr  6552