Theorem genpmu 7517

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 7474	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmu 7477	. . . 4 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑓 ∈ Q 𝑓 ∈ (2nd ‘𝐴))
3		rexex 2523	. . . 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 276	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑓 𝑓 ∈ (2nd ‘𝐴))
6		prop 7474	. . . . 5 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
7		prmu 7477	. . . . 5 ⊢ (⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P → ∃𝑔 ∈ Q 𝑔 ∈ (2nd ‘𝐵))
8		rexex 2523	. . . . 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 489	. . 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 7514	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵)) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
14	13	imp 124	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵))) → (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵)))
15		elprnqu 7481	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
16	1, 15	sylan 283	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) → 𝑓 ∈ Q)
17		elprnqu 7481	. . . . . . . . . 10 ⊢ ((⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵)) → 𝑔 ∈ Q)
18	6, 17	sylan 283	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵)) → 𝑔 ∈ Q)
19	16, 18	anim12i 338	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (2nd ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
20	19	an4s 588	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
21	12	caovcl 6029	. . . . . . 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 110	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → 𝑞 = (𝑓𝐺𝑔))
24	23	eleq1d 2246	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
25	22, 24	rspcedv 2846	. . . . 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 400	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (2nd ‘𝐴)) ∧ 𝑔 ∈ (2nd ‘𝐵)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
28	10, 27	exlimddv 1898	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (2nd ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
29	5, 28	exlimddv 1898	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 978   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∃wrex 2456  {crab 2459  ⟨cop 3596  ‘cfv 5217  (class class class)co 5875   ∈ cmpo 5877  1^st c1st 6139  2^nd c2nd 6140  Qcnq 7279  Pcnp 7290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-qs 6541  df-ni 7303  df-nqqs 7347  df-inp 7465

This theorem is referenced by:  addclpr  7536  mulclpr  7571