dicelvalN - Mathbox for Norm Megill

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Theorem dicelvalN 41506

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 41504	. . 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 7945	. . . . 5 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → (1^st ‘𝑌) = 𝑓)
13	10, 11	op2ndd 7946	. . . . . 6 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → (2^nd ‘𝑌) = 𝑠)
14	13	fveq1d 6837	. . . . 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 633	. . 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 5758	. 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 287	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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3441 ⟨cop 4587 class class class wbr 5099 {copab 5161 × cxp 5623 ‘cfv 6493 ℩crio 7316 1^st c1st 7933 2^nd c2nd 7934 lecple 17188 occoc 17189 Atomscatm 39591 LHypclh 40312 LTrncltrn 40429 TEndoctendo 41080 DIsoCcdic 41500

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-1st 7935 df-2nd 7936 df-dic 41501

This theorem is referenced by: dicelval2N 41510

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