genpelvl - Intuitionistic Logic Explorer

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Theorem genpelvl 7579

Description: Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-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
genpelvl	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝐶 = (𝑔𝐺ℎ)))

Distinct variable groups: 𝑥,𝑦,𝑧,𝑔,ℎ,𝑤,𝑣,𝐴 𝑥,𝐵,𝑦,𝑧,𝑔,ℎ,𝑤,𝑣 𝑥,𝐺,𝑦,𝑧,𝑔,ℎ,𝑤,𝑣 𝑔,𝐹 𝐶,𝑔,ℎ Allowed substitution hints: 𝐶(𝑥,𝑦,𝑧,𝑤,𝑣) 𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,ℎ)

Proof of Theorem genpelvl Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		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))}⟩)
2		genpelvl.2	. . . . . . 7 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
3	1, 2	genipv 7576	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) = ⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}⟩)
4	3	fveq2d 5562	. . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (1^st ‘(𝐴𝐹𝐵)) = (1^st ‘⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}⟩))
5		nqex 7430	. . . . . . 7 ⊢ Q ∈ V
6	5	rabex 4177	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
7	5	rabex 4177	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
8	6, 7	op1st 6204	. . . . 5 ⊢ (1^st ‘⟨{𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)}, {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)}⟩) = {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)}
9	4, 8	eqtrdi 2245	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (1^st ‘(𝐴𝐹𝐵)) = {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)})
10	9	eleq2d 2266	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)}))
11		elrabi 2917	. . 3 ⊢ (𝐶 ∈ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)} → 𝐶 ∈ Q)
12	10, 11	biimtrdi 163	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (1^st ‘(𝐴𝐹𝐵)) → 𝐶 ∈ Q))
13		prop 7542	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
14		elprnql 7548	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) → 𝑔 ∈ Q)
15	13, 14	sylan 283	. . . . . 6 ⊢ ((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) → 𝑔 ∈ Q)
16		prop 7542	. . . . . . 7 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
17		elprnql 7548	. . . . . . 7 ⊢ ((⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P ∧ ℎ ∈ (1^st ‘𝐵)) → ℎ ∈ Q)
18	16, 17	sylan 283	. . . . . 6 ⊢ ((𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵)) → ℎ ∈ Q)
19	2	caovcl 6078	. . . . . 6 ⊢ ((𝑔 ∈ Q ∧ ℎ ∈ Q) → (𝑔𝐺ℎ) ∈ Q)
20	15, 18, 19	syl2an 289	. . . . 5 ⊢ (((𝐴 ∈ P ∧ 𝑔 ∈ (1^st ‘𝐴)) ∧ (𝐵 ∈ P ∧ ℎ ∈ (1^st ‘𝐵))) → (𝑔𝐺ℎ) ∈ Q)
21	20	an4s 588	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ (1^st ‘𝐴) ∧ ℎ ∈ (1^st ‘𝐵))) → (𝑔𝐺ℎ) ∈ Q)
22		eleq1 2259	. . . 4 ⊢ (𝐶 = (𝑔𝐺ℎ) → (𝐶 ∈ Q ↔ (𝑔𝐺ℎ) ∈ Q))
23	21, 22	syl5ibrcom 157	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ (1^st ‘𝐴) ∧ ℎ ∈ (1^st ‘𝐵))) → (𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ Q))
24	23	rexlimdvva 2622	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ Q))
25		eqeq1 2203	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺ℎ) ↔ 𝐶 = (𝑔𝐺ℎ)))
26	25	2rexbidv 2522	. . . . 5 ⊢ (𝑓 = 𝐶 → (∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
27	26	elrab3 2921	. . . 4 ⊢ (𝐶 ∈ Q → (𝐶 ∈ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
28	10, 27	sylan9bb 462	. . 3 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐶 ∈ Q) → (𝐶 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝐶 = (𝑔𝐺ℎ)))
29	28	ex 115	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ Q → (𝐶 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝐶 = (𝑔𝐺ℎ))))
30	12, 24, 29	pm5.21ndd 706	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝐶 = (𝑔𝐺ℎ)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ 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: genpprecll 7581 genpcdl 7586 genprndl 7588 genpdisj 7590 genpassl 7591 addnqprlemrl 7624 mulnqprlemrl 7640 distrlem1prl 7649 distrlem5prl 7653 1idprl 7657 ltexprlemfl 7676 recexprlem1ssl 7700 recexprlemss1l 7702 cauappcvgprlemladdfl 7722

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