genpprecll - Intuitionistic Logic Explorer

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

Theorem genpprecll 7576

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 2193	. . 3 ⊢ (𝐶𝐺𝐷) = (𝐶𝐺𝐷)
2		rspceov 5961	. . 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 1337	. 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 7574	. 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → ((𝐶𝐺𝐷) ∈ (1^st ‘(𝐴𝐹𝐵)) ↔ ∃𝑔 ∈ (1^st ‘𝐴)∃ℎ ∈ (1^st ‘𝐵)(𝐶𝐺𝐷) = (𝑔𝐺ℎ)))
7	3, 6	imbitrrid 156	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 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 ∃wrex 2473 {crab 2476 ⟨cop 3622 ‘cfv 5255 (class class class)co 5919 ∈ cmpo 5921 1^st c1st 6193 2^nd c2nd 6194 Qcnq 7342 Pcnp 7353

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-id 4325 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-qs 6595 df-ni 7366 df-nqqs 7410 df-inp 7528

This theorem is referenced by: genpml 7579 genprndl 7583 addnqprl 7591 mulnqprl 7630 distrlem1prl 7644 distrlem4prl 7646 ltaddpr 7659 ltexprlemrl 7672 addcanprleml 7676 addcanprlemu 7677

W3C validator