Theorem genpmu 7319

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 7276	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmu 7279	. . . 4 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑓 ∈ Q 𝑓 ∈ (2nd ‘𝐴))
3		rexex 2477	. . . 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 274	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑓 𝑓 ∈ (2nd ‘𝐴))
6		prop 7276	. . . . 5 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
7		prmu 7279	. . . . 5 ⊢ (⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P → ∃𝑔 ∈ Q 𝑔 ∈ (2nd ‘𝐵))
8		rexex 2477	. . . . 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 480	. . 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 7316	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵)) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
14	13	imp 123	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵))) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵)))
15		elprnqu 7283	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
16	1, 15	sylan 281	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
17		elprnqu 7283	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵)) → 𝑔 ∈ Q)
18	6, 17	sylan 281	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵)) → 𝑔 ∈ Q)
19	16, 18	anim12i 336	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
20	19	an4s 577	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
21	12	caovcl 5918	. . . . . . 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 109	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → 𝑞 = (𝑓𝐺𝑔))
24	23	eleq1d 2206	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
25	22, 24	rspcedv 2788	. . . . 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 397	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ 𝑔 ∈ (2nd ‘𝐵)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
28	10, 27	exlimddv 1870	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (2nd ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
29	5, 28	exlimddv 1870	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2415  {crab 2418  ⟨cop 3525  ‘cfv 5118  (class class class)co 5767   ∈ cmpo 5769  1^st c1st 6029  2^nd c2nd 6030  Qcnq 7081  Pcnp 7092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-qs 6428  df-ni 7105  df-nqqs 7149  df-inp 7267

This theorem is referenced by:  addclpr  7338  mulclpr  7373