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Theorem genpml 7584

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 7542	. . . 4 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
2		prml 7544	. . . 4 ⊢ (⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P → ∃𝑓 ∈ Q 𝑓 ∈ (1^st ‘𝐴))
3		rexex 2543	. . . 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 7542	. . . . 5 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
7		prml 7544	. . . . 5 ⊢ (⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P → ∃𝑔 ∈ Q 𝑔 ∈ (1^st ‘𝐵))
8		rexex 2543	. . . . 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 7581	. . . . . 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 7548	. . . . . . . . . 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 7548	. . . . . . . . . 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 588	. . . . . . 7 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑓 ∈ (1^st ‘𝐴) ∧ 𝑔 ∈ (1^st ‘𝐵))) → (𝑓 ∈ Q ∧ 𝑔 ∈ Q))
21	12	caovcl 6078	. . . . . . 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 2265	. . . . . 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 2872	. . . . 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 1913	. 2 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝑓 ∈ (1^st ‘𝐴)) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))
29	5, 28	exlimddv 1913	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ∃𝑞 ∈ Q 𝑞 ∈ (1^st ‘(𝐴𝐹𝐵)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∃wex 1506 ∈ wcel 2167 ∃wrex 2476 {crab 2479 ⟨cop 3625 ‘cfv 5258 (class class class)co 5922 ∈ cmpo 5924 1^st c1st 6196 2^nd c2nd 6197 Qcnq 7347 Pcnp 7358

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 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-qs 6598 df-ni 7371 df-nqqs 7415 df-inp 7533

This theorem is referenced by: addclpr 7604 mulclpr 7639

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