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Theorem genpml 7226

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 7184	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prml 7186	. . . 4 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑓 ∈ Q 𝑓 ∈ (1^st ‘𝐴))
3		rexex 2438	. . . 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 272	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑓 𝑓 ∈ (1^st ‘𝐴))
6		prop 7184	. . . . 5 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
7		prml 7186	. . . . 5 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P → ∃𝑔 ∈ Q 𝑔 ∈ (1^st ‘𝐵))
8		rexex 2438	. . . . 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 476	. . 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 7223	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵)) → (𝑓𝐺𝑔) ∈ (1^st ‘(𝐴𝐹𝐵))))
14	13	imp 123	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) → (𝑓𝐺𝑔) ∈ (1^st ‘(𝐴𝐹𝐵)))
15		elprnql 7190	. . . . . . . . . 10 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
16	1, 15	sylan 279	. . . . . . . . 9 ⊢ ((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) → 𝑓 ∈ Q)
17		elprnql 7190	. . . . . . . . . 10 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵)) → 𝑔 ∈ Q)
18	6, 17	sylan 279	. . . . . . . . 9 ⊢ ((𝐵 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵)) → 𝑔 ∈ Q)
19	16, 18	anim12i 334	. . . . . . . 8 ⊢ (((𝐴 ∈ P ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
20	19	an4s 558	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
21	12	caovcl 5857	. . . . . . 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 109	. . . . . . 7 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) ∧ 𝑞 = (𝑓𝐺𝑔)) → 𝑞 = (𝑓𝐺𝑔))
24	23	eleq1d 2168	. . . . . 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 2748	. . . . 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 395	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ 𝑔 ∈ (1^st ‘𝐵)) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))
28	10, 27	exlimddv 1837	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))
29	5, 28	exlimddv 1837	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 930 = wceq 1299 ∃wex 1436 ∈ wcel 1448 ∃wrex 2376 {crab 2379 ⟨cop 3477 ‘cfv 5059 (class class class)co 5706 ∈ cmpo 5708 1^st c1st 5967 2^nd c2nd 5968 Qcnq 6989 Pcnp 7000

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-coll 3983 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-iinf 4440

This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-ral 2380 df-rex 2381 df-reu 2382 df-rab 2384 df-v 2643 df-sbc 2863 df-csb 2956 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-int 3719 df-iun 3762 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-iom 4443 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-f1 5064 df-fo 5065 df-f1o 5066 df-fv 5067 df-ov 5709 df-oprab 5710 df-mpo 5711 df-1st 5969 df-2nd 5970 df-qs 6365 df-ni 7013 df-nqqs 7057 df-inp 7175

This theorem is referenced by: addclpr 7246 mulclpr 7281

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