genpelvl - Intuitionistic Logic Explorer

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

Theorem genpelvl 6360

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 6357	. . . . . 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 5103	. . . . 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 6216	. . . . . . 7 ⊢ Q ∈ V
6	5	rabex 3871	. . . . . 6 ⊢ {f ∈ Q ∣ ∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)f = (g𝐺ℎ)} ∈ V
7	5	rabex 3871	. . . . . 6 ⊢ {f ∈ Q ∣ ∃g ∈ (2^nd ‘A)∃ℎ ∈ (2^nd ‘B)f = (g𝐺ℎ)} ∈ V
8	6, 7	op1st 5692	. . . . 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 2066	. . . 4 ⊢ ((A ∈ P ∧ B ∈ P) → (1^st ‘(A𝐹B)) = {f ∈ Q ∣ ∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)f = (g𝐺ℎ)})
10	9	eleq2d 2085	. . 3 ⊢ ((A ∈ P ∧ B ∈ P) → (𝐶 ∈ (1^st ‘(A𝐹B)) ↔ 𝐶 ∈ {f ∈ Q ∣ ∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)f = (g𝐺ℎ)}))
11		elrabi 2668	. . 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 6323	. . . . . . 7 ⊢ (A ∈ P → ⟨(1^st ‘A), (2^nd ‘A)⟩ ∈ P)
14		elprnql 6329	. . . . . . 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 6323	. . . . . . 7 ⊢ (B ∈ P → ⟨(1^st ‘B), (2^nd ‘B)⟩ ∈ P)
17		elprnql 6329	. . . . . . 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 5574	. . . . . 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 509	. . . 4 ⊢ (((A ∈ P ∧ B ∈ P) ∧ (g ∈ (1^st ‘A) ∧ ℎ ∈ (1^st ‘B))) → (g𝐺ℎ) ∈ Q)
22		eleq1 2078	. . . 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 2414	. 2 ⊢ ((A ∈ P ∧ B ∈ P) → (∃g ∈ (1^st ‘A)∃ℎ ∈ (1^st ‘B)𝐶 = (g𝐺ℎ) → 𝐶 ∈ Q))
25		eqeq1 2024	. . . . . 6 ⊢ (f = 𝐶 → (f = (g𝐺ℎ) ↔ 𝐶 = (g𝐺ℎ)))
26	25	2rexbidv 2323	. . . . 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 2672	. . . 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 438	. . 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 608	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 871 = wceq 1226 ∈ wcel 1370 ∃wrex 2281 {crab 2284 ⟨cop 3349 ‘cfv 4825 (class class class)co 5432 ↦ cmpt2 5434 1^st c1st 5684 2^nd c2nd 5685 Qcnq 6134 Pcnp 6145

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 532 ax-in2 533 ax-io 617 ax-5 1312 ax-7 1313 ax-gen 1314 ax-ie1 1359 ax-ie2 1360 ax-8 1372 ax-10 1373 ax-11 1374 ax-i12 1375 ax-bnd 1376 ax-4 1377 ax-13 1381 ax-14 1382 ax-17 1396 ax-i9 1400 ax-ial 1405 ax-i5r 1406 ax-ext 2000 ax-coll 3842 ax-sep 3845 ax-pow 3897 ax-pr 3914 ax-un 4116 ax-setind 4200 ax-iinf 4234

This theorem depends on definitions: df-bi 110 df-3an 873 df-tru 1229 df-fal 1232 df-nf 1326 df-sb 1624 df-eu 1881 df-mo 1882 df-clab 2005 df-cleq 2011 df-clel 2014 df-nfc 2145 df-ne 2184 df-ral 2285 df-rex 2286 df-reu 2287 df-rab 2289 df-v 2533 df-sbc 2738 df-csb 2826 df-dif 2893 df-un 2895 df-in 2897 df-ss 2904 df-pw 3332 df-sn 3352 df-pr 3353 df-op 3355 df-uni 3551 df-int 3586 df-iun 3629 df-br 3735 df-opab 3789 df-mpt 3790 df-id 4000 df-iom 4237 df-xp 4274 df-rel 4275 df-cnv 4276 df-co 4277 df-dm 4278 df-rn 4279 df-res 4280 df-ima 4281 df-iota 4790 df-fun 4827 df-fn 4828 df-f 4829 df-f1 4830 df-fo 4831 df-f1o 4832 df-fv 4833 df-ov 5435 df-oprab 5436 df-mpt2 5437 df-1st 5686 df-2nd 5687 df-qs 6019 df-ni 6158 df-nqqs 6201 df-inp 6314

This theorem is referenced by: genpprecll 6362 genpcdl 6368 genprndl 6370 genpdisj 6372 genpassl 6373 distrlem1prl 6415 distrlem5prl 6419 1idprl 6423 ltexprlemfl 6440 recexprlem1ssl 6461 recexprlemss1l 6463

W3C validator