genpprecll - Intuitionistic Logic Explorer

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

Theorem genpprecll 6669

Description: Pre-closure law for general operation on lower cuts. (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
genpprecll	⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶 ∈ (1^st ‘𝐴) ∧ 𝐷 ∈ (1^st ‘𝐵)) → (𝐶𝐺𝐷) ∈ (1^st ‘(𝐴𝐹𝐵))))

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

Proof of Theorem genpprecll Dummy variables 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2056	. . 3 ⊢ (𝐶𝐺𝐷) = (𝐶𝐺𝐷)
2		rspceov 5574	. . 3 ⊢ ((𝐶 ∈ (1^st ‘𝐴) ∧ 𝐷 ∈ (1^st ‘𝐵) ∧ (𝐶𝐺𝐷) = (𝐶𝐺𝐷)) → ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)(𝐶𝐺𝐷) = (𝑔𝐺ℎ))
3	1, 2	mp3an3 1232	. 2 ⊢ ((𝐶 ∈ (1^st ‘𝐴) ∧ 𝐷 ∈ (1^st ‘𝐵)) → ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)(𝐶𝐺𝐷) = (𝑔𝐺ℎ))
4		genpelvl.1	. . 3 ⊢ 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))}⟩)
5		genpelvl.2	. . 3 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q)
6	4, 5	genpelvl 6667	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶𝐺𝐷) ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)(𝐶𝐺𝐷) = (𝑔𝐺ℎ)))
7	3, 6	syl5ibr 149	1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶 ∈ (1^st ‘𝐴) ∧ 𝐷 ∈ (1^st ‘𝐵)) → (𝐶𝐺𝐷) ∈ (1^st ‘(𝐴𝐹𝐵))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 101 ∧ w3a 896 = wceq 1259 ∈ wcel 1409 ∃wrex 2324 {crab 2327 ⟨cop 3405 ‘cfv 4929 (class class class)co 5539 ↦ cmpt2 5541 1^st c1st 5792 2^nd c2nd 5793 Qcnq 6435 Pcnp 6446

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 103 ax-ia2 104 ax-ia3 105 ax-in1 554 ax-in2 555 ax-io 640 ax-5 1352 ax-7 1353 ax-gen 1354 ax-ie1 1398 ax-ie2 1399 ax-8 1411 ax-10 1412 ax-11 1413 ax-i12 1414 ax-bndl 1415 ax-4 1416 ax-13 1420 ax-14 1421 ax-17 1435 ax-i9 1439 ax-ial 1443 ax-i5r 1444 ax-ext 2038 ax-coll 3899 ax-sep 3902 ax-pow 3954 ax-pr 3971 ax-un 4197 ax-setind 4289 ax-iinf 4338

This theorem depends on definitions: df-bi 114 df-3an 898 df-tru 1262 df-fal 1265 df-nf 1366 df-sb 1662 df-eu 1919 df-mo 1920 df-clab 2043 df-cleq 2049 df-clel 2052 df-nfc 2183 df-ne 2221 df-ral 2328 df-rex 2329 df-reu 2330 df-rab 2332 df-v 2576 df-sbc 2787 df-csb 2880 df-dif 2947 df-un 2949 df-in 2951 df-ss 2958 df-pw 3388 df-sn 3408 df-pr 3409 df-op 3411 df-uni 3608 df-int 3643 df-iun 3686 df-br 3792 df-opab 3846 df-mpt 3847 df-id 4057 df-iom 4341 df-xp 4378 df-rel 4379 df-cnv 4380 df-co 4381 df-dm 4382 df-rn 4383 df-res 4384 df-ima 4385 df-iota 4894 df-fun 4931 df-fn 4932 df-f 4933 df-f1 4934 df-fo 4935 df-f1o 4936 df-fv 4937 df-ov 5542 df-oprab 5543 df-mpt2 5544 df-1st 5794 df-2nd 5795 df-qs 6142 df-ni 6459 df-nqqs 6503 df-inp 6621

This theorem is referenced by: genpml 6672 genprndl 6676 addnqprl 6684 mulnqprl 6723 distrlem1prl 6737 distrlem4prl 6739 ltaddpr 6752 ltexprlemrl 6765 addcanprleml 6769 addcanprlemu 6770

W3C validator