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Theorem genpml 7885

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 7843	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prml 7845	. . . 4 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑓 ∈ Q 𝑓 ∈ (1^st ‘𝐴))
3		rexex 2596	. . . 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 7843	. . . . 5 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
7		prml 7845	. . . . 5 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P → ∃𝑔 ∈ Q 𝑔 ∈ (1^st ‘𝐵))
8		rexex 2596	. . . . 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 493	. . 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 7882	. . . . . 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 7849	. . . . . . . . . 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 7849	. . . . . . . . . 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 596	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
21	12	caovcl 6244	. . . . . . 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 2307	. . . . . 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 2933	. . . . 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 404	. . 3 ⊢ ((((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) ∧ 𝑔 ∈ (1^st ‘𝐵)) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))
28	10, 27	exlimddv 1954	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))
29	5, 28	exlimddv 1954	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ∧ w3a 1009 = wceq 1402 ∃wex 1545 ∈ wcel 2209 ∃wrex 2529 {crab 2532 ⟨cop 3712 ‘cfv 5377 (class class class)co 6085 ∈ cmpo 6087 1^st c1st 6372 2^nd c2nd 6373 Qcnq 7648 Pcnp 7659

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-id 4438 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-qs 6813 df-ni 7672 df-nqqs 7716 df-inp 7834

This theorem is used by: addclpr 7905 mulclpr 7940

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