Theorem genpmu 7651

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 7608	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P)
2		prmu 7611	. . . 4 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ P → ∃𝑓 ∈ Q 𝑓 ∈ (2nd ‘𝐴))
3		rexex 2553	. . . 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 7608	. . . . 5 ⊢ (𝐵 ∈ P → ⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P)
7		prmu 7611	. . . . 5 ⊢ (⟨(1st ‘𝐵), (2nd ‘𝐵)⟩ ∈ P → ∃𝑔 ∈ Q 𝑔 ∈ (2nd ‘𝐵))
8		rexex 2553	. . . . 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 7648	. . . . . 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 7615	. . . . . . . . . 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 7615	. . . . . . . . . 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 6114	. . . . . . 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 2275	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (2nd ‘𝐴) ∧ 𝑔 ∈ (2nd ‘𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (2nd ‘(𝐴𝐹𝐵))))
25	22, 24	rspcedv 2885	. . . . 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 1923	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (2nd ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))
29	5, 28	exlimddv 1923	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (2nd ‘(𝐴𝐹𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373  ∃wex 1516   ∈ wcel 2177  ∃wrex 2486  {crab 2489  ⟨cop 3641  ‘cfv 5280  (class class class)co 5957   ∈ cmpo 5959  1^st c1st 6237  2^nd c2nd 6238  Qcnq 7413  Pcnp 7424

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-qs 6639  df-ni 7437  df-nqqs 7481  df-inp 7599

This theorem is referenced by:  addclpr  7670  mulclpr  7705