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Theorem genpml 7700

Description: The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.)

**Hypotheses**
Ref	Expression
genpelvl.1	⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
genpelvl.2	⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)

**Assertion**
Ref	Expression
genpml	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(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 genpml Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prop 7658	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prml 7660	. . . 4 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑓 ∈ Q 𝑓 ∈ (1^st ‘𝐴))
3		rexex 2576	. . . 4 ⊢ (∃𝑓 ∈ Q 𝑓 ∈ (1^st ‘𝐴) → ∃𝑓 𝑓 ∈ (1^st ‘𝐴))
4	1, 2, 3	3syl 17	. . 3 ⊢ (𝐴 ∈ P → ∃𝑓 𝑓 ∈ (1^st ‘𝐴))
5	4	adantr 276	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑓 𝑓 ∈ (1^st ‘𝐴))
6		prop 7658	. . . . 5 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
7		prml 7660	. . . . 5 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P → ∃𝑔 ∈ Q 𝑔 ∈ (1^st ‘𝐵))
8		rexex 2576	. . . . 5 ⊢ (∃𝑔 ∈ Q 𝑔 ∈ (1^st ‘𝐵) → ∃𝑔 𝑔 ∈ (1^st ‘𝐵))
9	6, 7, 8	3syl 17	. . . 4 ⊢ (𝐵 ∈ P → ∃𝑔 𝑔 ∈ (1^st ‘𝐵))
10	9	ad2antlr 489	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) → ∃𝑔 𝑔 ∈ (1^st ‘𝐵))
11		genpelvl.1	. . . . . . 7 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ ⟨{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1^st ‘𝑤) ∧ 𝑧 ∈ (1^st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2^nd ‘𝑤) ∧ 𝑧 ∈ (2^nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}⟩)
12		genpelvl.2	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
13	11, 12	genpprecll 7697	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵)) → (𝑓𝐺𝑔) ∈ (1^st ‘(𝐴𝐹𝐵))))
14	13	imp 124	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) → (𝑓𝐺𝑔) ∈ (1^st ‘(𝐴𝐹𝐵)))
15		elprnql 7664	. . . . . . . . . 10 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
16	1, 15	sylan 283	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
17		elprnql 7664	. . . . . . . . . 10 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵)) → 𝑔 ∈ Q)
18	6, 17	sylan 283	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵)) → 𝑔 ∈ Q)
19	16, 18	anim12i 338	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
20	19	an4s 590	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
21	12	caovcl 6159	. . . . . . 7 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q) → (𝑓𝐺𝑔) ∈ Q)
22	20, 21	syl 14	. . . . . 6 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) → (𝑓𝐺𝑔) ∈ Q)
23		simpr 110	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → 𝑞 = (𝑓𝐺𝑔))
24	23	eleq1d 2298	. . . . . 6 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → (𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ (𝑓𝐺𝑔) ∈ (1^st ‘(𝐴𝐹𝐵))))
25	22, 24	rspcedv 2911	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) → ((𝑓𝐺𝑔) ∈ (1^st ‘(𝐴𝐹𝐵)) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵))))
26	14, 25	mpd 13	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))
27	26	anassrs 400	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ 𝑔 ∈ (1^st ‘𝐵)) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))
28	10, 27	exlimddv 1945	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))
29	5, 28	exlimddv 1945	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1002 = wceq 1395 ∃wex 1538 ∈ wcel 2200 ∃wrex 2509 {crab 2512 ⟨cop 3669 ‘cfv 5317 (class class class)co 6000 ∈ cmpo 6002 1^st c1st 6282 2^nd c2nd 6283 Qcnq 7463 Pcnp 7474

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-id 4383 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-qs 6684 df-ni 7487 df-nqqs 7531 df-inp 7649

This theorem is referenced by: addclpr 7720 mulclpr 7755

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