dicelvalN - Mathbox for Norm Megill

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Theorem dicelvalN 41583

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 41581	. . 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 3446	. . . . . 6 ⊢ 𝑓 ∈ V
11		vex 3446	. . . . . 6 ⊢ 𝑠 ∈ V
12	10, 11	op1std 7955	. . . . 5 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → (1^st ‘𝑌) = 𝑓)
13	10, 11	op2ndd 7956	. . . . . 6 ⊢ (𝑌 = ⟨𝑓, 𝑠⟩ → (2^nd ‘𝑌) = 𝑠)
14	13	fveq1d 6846	. . . . 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 5767	. 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 3442 ⟨cop 4588 class class class wbr 5100 {copab 5162 × cxp 5632 ‘cfv 6502 ℩crio 7326 1^st c1st 7943 2^nd c2nd 7944 lecple 17198 occoc 17199 Atomscatm 39668 LHypclh 40389 LTrncltrn 40506 TEndoctendo 41157 DIsoCcdic 41577

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 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692

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 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5529 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-1st 7945 df-2nd 7946 df-dic 41578

This theorem is referenced by: dicelval2N 41587

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