genpelvl - Intuitionistic Logic Explorer

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Theorem genpelvl 6495

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 6492	. . . . . 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 5125	. . . . 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 6347	. . . . . . 7 ⊢ Q ∈ V
6	5	rabex 3892	. . . . . 6 ⊢ {f ∈ Q ∣ ∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)f = (g𝐺ℎ)} ∈ V
7	5	rabex 3892	. . . . . 6 ⊢ {f ∈ Q ∣ ∃g ∈ (2^nd ‘A)∃ℎ ∈ (2^nd ‘B)f = (g𝐺ℎ)} ∈ V
8	6, 7	op1st 5715	. . . . 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	syl6eq 2085	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P) → (1^st ‘(A𝐹B)) = {f ∈ Q ∣ ∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)f = (g𝐺ℎ)})
10	9	eleq2d 2104	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → (𝐶 ∈ (1^st ‘(A𝐹B)) ↔ 𝐶 ∈ {f ∈ Q ∣ ∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)f = (g𝐺ℎ)}))
11		elrabi 2689	. . 3 ⊢ (𝐶 ∈ {f ∈ Q ∣ ∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)f = (g𝐺ℎ)} → 𝐶 ∈ Q)
12	10, 11	syl6bi 152	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → (𝐶 ∈ (1^st ‘(A𝐹B)) → 𝐶 ∈ Q))
13		prop 6458	. . . . . . 7 ⊢ (A ∈ P → ⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P)
14		elprnql 6464	. . . . . . 7 ⊢ ((⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P ∧ g ∈ (1^st ‘A)) → g ∈ Q)
15	13, 14	sylan 267	. . . . . 6 ⊢ ((A ∈ P ∧ g ∈ (1^st ‘A)) → g ∈ Q)
16		prop 6458	. . . . . . 7 ⊢ (B ∈ P → ⟨(1^st ‘B), (2^nd ‘B)⟩ ∈ P)
17		elprnql 6464	. . . . . . 7 ⊢ ((⟨(1^st ‘B), (2^nd ‘B)⟩ ∈ P ∧ ℎ ∈ (1^st ‘B)) → ℎ ∈ Q)
18	16, 17	sylan 267	. . . . . 6 ⊢ ((B ∈ P ∧ ℎ ∈ (1^st ‘B)) → ℎ ∈ Q)
19	2	caovcl 5597	. . . . . 6 ⊢ ((g ∈ Q ∧ ℎ ∈ Q) → (g𝐺ℎ) ∈ Q)
20	15, 18, 19	syl2an 273	. . . . 5 ⊢ (((A ∈ P ∧ g ∈ (1^st ‘A)) ∧ (B ∈ P ∧ ℎ ∈ (1^st ‘B))) → (g𝐺ℎ) ∈ Q)
21	20	an4s 522	. . . 4 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (g ∈ (1^st ‘A) ∧ ℎ ∈ (1^st ‘B))) → (g𝐺ℎ) ∈ Q)
22		eleq1 2097	. . . 4 ⊢ (𝐶 = (g𝐺ℎ) → (𝐶 ∈ Q ↔ (g𝐺ℎ) ∈ Q))
23	21, 22	syl5ibrcom 146	. . 3 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (g ∈ (1^st ‘A) ∧ ℎ ∈ (1^st ‘B))) → (𝐶 = (g𝐺ℎ) → 𝐶 ∈ Q))
24	23	rexlimdvva 2434	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → (∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)𝐶 = (g𝐺ℎ) → 𝐶 ∈ Q))
25		eqeq1 2043	. . . . . 6 ⊢ (f = 𝐶 → (f = (g𝐺ℎ) ↔ 𝐶 = (g𝐺ℎ)))
26	25	2rexbidv 2343	. . . . 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 2693	. . . 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 435	. . 3 ⊢ (((A ∈ P ∧ B ∈ P) ∧ 𝐶 ∈ Q) → (𝐶 ∈ (1^st ‘(A𝐹B)) ↔ ∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)𝐶 = (g𝐺ℎ)))
29	28	ex 108	. 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 620	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 97 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 ∃wrex 2301 {crab 2304 ⟨cop 3370 ‘cfv 4845 (class class class)co 5455 ↦ cmpt2 5457 1^st c1st 5707 2^nd c2nd 5708 Qcnq 6264 Pcnp 6275

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254

This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-qs 6048 df-ni 6288 df-nqqs 6332 df-inp 6449

This theorem is referenced by: genpprecll 6497 genpcdl 6502 genprndl 6504 genpdisj 6506 genpassl 6507 addnqprlemrl 6538 distrlem1prl 6558 distrlem5prl 6562 1idprl 6566 ltexprlemfl 6583 recexprlem1ssl 6605 recexprlemss1l 6607 cauappcvgprlemladdfl 6627

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