Theorem genpmu 6367

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 6323	. . . 4 ⊢ (A ∈ P → ⟨(1st ‘A), (2nd ‘A)⟩ ∈ P)
2		prmu 6326	. . . 4 ⊢ (⟨(1st ‘A), (2nd ‘A)⟩ ∈ P → ∃f ∈ Q f ∈ (2nd ‘A))
3		rexex 2342	. . . 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 6323	. . . . 5 ⊢ (B ∈ P → ⟨(1st ‘B), (2nd ‘B)⟩ ∈ P)
7		prmu 6326	. . . . 5 ⊢ (⟨(1st ‘B), (2nd ‘B)⟩ ∈ P → ∃g ∈ Q g ∈ (2nd ‘B))
8		rexex 2342	. . . . 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 462	. . 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 6363	. . . . . 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 6330	. . . . . . . . . 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 6330	. . . . . . . . . 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 509	. . . . . . 7 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B))) → (f ∈ Q ∧ g ∈ Q))
21	12	caovcl 5574	. . . . . . 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 2084	. . . . . 6 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ (f ∈ (2nd ‘A) ∧ g ∈ (2nd ‘B))) ∧ 𝑞 = (f𝐺g)) → (𝑞 ∈ (2nd ‘(A𝐹B)) ↔ (f𝐺g) ∈ (2nd ‘(A𝐹B))))
25	22, 24	rspcedv 2633	. . . . 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 382	. . 3 ⊢ ((((A ∈ P ∧ B ∈ P) ∧ f ∈ (2nd ‘A)) ∧ g ∈ (2nd ‘B)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B)))
28	10, 27	exlimddv 1756	. 2 ⊢ (((A ∈ P ∧ B ∈ P) ∧ f ∈ (2nd ‘A)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B)))
29	5, 28	exlimddv 1756	1 ⊢ ((A ∈ P ∧ B ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(A𝐹B)))

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 871   = wceq 1226  ∃wex 1358   ∈ wcel 1370  ∃wrex 2281  {crab 2284  ⟨cop 3349  ‘cfv 4825  (class class class)co 5432   ↦ cmpt2 5434  1^st c1st 5684  2^nd c2nd 5685  Qcnq 6134  Pcnp 6145

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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200  ax-iinf 4234

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-int 3586  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-iom 4237  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-1st 5686  df-2nd 5687  df-qs 6019  df-ni 6158  df-nqqs 6201  df-inp 6314

This theorem is referenced by:  addclpr  6386  mulclpr  6410