genpelvl - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > genpelvl		GIF version

Theorem genpelvl 7792

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 7789	. . . . . 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 5652	. . . . 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 7643	. . . . . . 7 ⊢ Q ∈ V
6	5	rabex 4239	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
7	5	rabex 4239	. . . . . 6 ⊢ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (2^nd ‘𝐴)∃ℎ ∈ (2^nd ‘𝐵)𝑓 = (𝑔𝐺ℎ)} ∈ V
8	6, 7	op1st 6318	. . . . 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 2280	. . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (1^st ‘(𝐴𝐹𝐵)) = {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)})
10	9	eleq2d 2301	. . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐶 ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ 𝐶 ∈ {𝑓 ∈ Q ∣ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝑓 = (𝑔𝐺ℎ)}))
11		elrabi 2960	. . 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 7755	. . . . . . 7 ⊢ (𝐴 ∈ P → ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∈ P)
14		elprnql 7761	. . . . . . 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 7755	. . . . . . 7 ⊢ (𝐵 ∈ P → ⟨(1^st ‘𝐵), (2^nd ‘𝐵)⟩ ∈ P)
17		elprnql 7761	. . . . . . 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 6187	. . . . . 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 592	. . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ (𝑔 ∈ (1^st ‘𝐴) ∧ ℎ ∈ (1^st ‘𝐵))) → (𝑔𝐺ℎ) ∈ Q)
22		eleq1 2294	. . . 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 2659	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)𝐶 = (𝑔𝐺ℎ) → 𝐶 ∈ Q))
25		eqeq1 2238	. . . . . 6 ⊢ (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺ℎ) ↔ 𝐶 = (𝑔𝐺ℎ)))
26	25	2rexbidv 2558	. . . . 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 2964	. . . 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 713	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 1005 = wceq 1398 ∈ wcel 2202 ∃wrex 2512 {crab 2515 ⟨cop 3676 ‘cfv 5333 (class class class)co 6028 ∈ cmpo 6030 1^st c1st 6310 2^nd c2nd 6311 Qcnq 7560 Pcnp 7571

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-id 4396 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-qs 6751 df-ni 7584 df-nqqs 7628 df-inp 7746

This theorem is referenced by: genpprecll 7794 genpcdl 7799 genprndl 7801 genpdisj 7803 genpassl 7804 addnqprlemrl 7837 mulnqprlemrl 7853 distrlem1prl 7862 distrlem5prl 7866 1idprl 7870 ltexprlemfl 7889 recexprlem1ssl 7913 recexprlemss1l 7915 cauappcvgprlemladdfl 7935

W3C validator