dicelvalN - Mathbox for Norm Megill

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Theorem dicelvalN 39413

Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
dicval.l	⊢ ≤ = (le‘𝐾)
dicval.a	⊢ 𝐴 = (Atoms‘𝐾)
dicval.h	⊢ 𝐻 = (LHyp‘𝐾)
dicval.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dicval.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicval.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dicval.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
dicelvalN	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1^st ‘𝑌) = ((2^nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2^nd ‘𝑌) ∈ 𝐸))))

Distinct variable groups: 𝑔,𝐾 𝑇,𝑔 𝑔,𝑊 𝑄,𝑔 Allowed substitution hints: 𝐴(𝑔) 𝑃(𝑔) 𝐸(𝑔) 𝐻(𝑔) 𝐼(𝑔) ≤ (𝑔) 𝑉(𝑔) 𝑌(𝑔)

Proof of Theorem dicelvalN Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dicval.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		dicval.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3		dicval.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		dicval.p	. . . 4 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
5		dicval.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6		dicval.e	. . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
7		dicval.i	. . . 4 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dicval 39411	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)})
9	8	eleq2d 2823	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)}))
10		vex 3445	. . . . . 6 ⊢ 𝑓 ∈ V
11		vex 3445	. . . . . 6 ⊢ 𝑠 ∈ V
12	10, 11	op1std 7888	. . . . 5 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → (1^st ‘𝑌) = 𝑓)
13	10, 11	op2ndd 7889	. . . . . 6 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → (2^nd ‘𝑌) = 𝑠)
14	13	fveq1d 6814	. . . . 5 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → ((2^nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
15	12, 14	eqeq12d 2753	. . . 4 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → ((1^st ‘𝑌) = ((2^nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ↔ 𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))))
16	13	eleq1d 2822	. . . 4 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → ((2^nd ‘𝑌) ∈ 𝐸 ↔ 𝑠 ∈ 𝐸))
17	15, 16	anbi12d 631	. . 3 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → (((1^st ‘𝑌) = ((2^nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2^nd ‘𝑌) ∈ 𝐸) ↔ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)))
18	17	elopaba 5738	. 2 ⊢ (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ 𝑠 ∈ 𝐸)} ↔ (𝑌 ∈ (V × V) ∧ ((1^st ‘𝑌) = ((2^nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2^nd ‘𝑌) ∈ 𝐸)))
19	9, 18	bitrdi 286	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1^st ‘𝑌) = ((2^nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2^nd ‘𝑌) ∈ 𝐸))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ⟨cop 4577 class class class wbr 5087 {copab 5149 × cxp 5606 ‘cfv 6466 ℩crio 7273 1^st c1st 7876 2^nd c2nd 7877 lecple 17046 occoc 17047 Atomscatm 37497 LHypclh 38219 LTrncltrn 38336 TEndoctendo 38987 DIsoCcdic 39407

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2708 ax-rep 5224 ax-sep 5238 ax-nul 5245 ax-pow 5303 ax-pr 5367 ax-un 7630

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3351 df-rab 3405 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4268 df-if 4472 df-pw 4547 df-sn 4572 df-pr 4574 df-op 4578 df-uni 4851 df-iun 4939 df-br 5088 df-opab 5150 df-mpt 5171 df-id 5507 df-xp 5614 df-rel 5615 df-cnv 5616 df-co 5617 df-dm 5618 df-rn 5619 df-res 5620 df-ima 5621 df-iota 6418 df-fun 6468 df-fn 6469 df-f 6470 df-f1 6471 df-fo 6472 df-f1o 6473 df-fv 6474 df-riota 7274 df-1st 7878 df-2nd 7879 df-dic 39408

This theorem is referenced by: dicelval2N 39417

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