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Theorem genpprecll 7845

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 2234	. . 3 ⊢ (𝐶𝐺𝐷) = (𝐶𝐺𝐷)
2		rspceov 6101	. . 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 1363	. 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 7843	. 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 1005 = wceq 1398 ∈ wcel 2205 ∃wrex 2523 {crab 2526 ⟨cop 3697 ‘cfv 5357 (class class class)co 6058 ∈ cmpo 6060 1^st c1st 6345 2^nd c2nd 6346 Qcnq 7611 Pcnp 7622

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 2207 ax-14 2208 ax-ext 2216 ax-coll 4230 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-reu 2529 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-iun 3998 df-br 4115 df-opab 4177 df-mpt 4178 df-id 4419 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-res 4766 df-ima 4767 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-f1 5362 df-fo 5363 df-f1o 5364 df-fv 5365 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1st 6347 df-2nd 6348 df-qs 6786 df-ni 7635 df-nqqs 7679 df-inp 7797

This theorem is referenced by: genpml 7848 genprndl 7852 addnqprl 7860 mulnqprl 7899 distrlem1prl 7913 distrlem4prl 7915 ltaddpr 7928 ltexprlemrl 7941 addcanprleml 7945 addcanprlemu 7946

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