Theorem dicelval2N 41806

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)
dicval2.g	⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)

Assertion

Ref	Expression
dicelval2N	⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ∧ (2nd ‘𝑌) ∈ 𝐸))))

Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,𝑊   𝑄,𝑔

Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝐸(𝑔)   𝐺(𝑔)   𝐻(𝑔)   𝐼(𝑔)   ≤ (𝑔)   𝑉(𝑔)   𝑌(𝑔)

Proof of Theorem dicelval2N

Step	Hyp	Ref	Expression
1		dicval.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		dicval.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		dicval.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		dicval.p	. . 3 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
5		dicval.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
6		dicval.e	. . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
7		dicval.i	. . 3 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
8	1, 2, 3, 4, 5, 6, 7	dicelvalN 41802	. 2 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸))))
9		dicval2.g	. . . . . 6 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
10	9	fveq2i 6870	. . . . 5 ⊢ ((2nd ‘𝑌)‘𝐺) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄))
11	10	eqeq2i 2775	. . . 4 ⊢ ((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ↔ (1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)))
12	11	anbi1i 633	. . 3 ⊢ (((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ∧ (2nd ‘𝑌) ∈ 𝐸) ↔ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸))
13	12	anbi2i 632	. 2 ⊢ ((𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ∧ (2nd ‘𝑌) ∈ 𝐸)) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘(℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)) ∧ (2nd ‘𝑌) ∈ 𝐸)))
14	8, 13	bitr4di 291	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st ‘𝑌) = ((2nd ‘𝑌)‘𝐺) ∧ (2nd ‘𝑌) ∈ 𝐸))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Vcvv 3454   class class class wbr 5100   × cxp 5645  ‘cfv 6521  ℩crio 7352  1^st c1st 7968  2^nd c2nd 7969  lecple 17293  occoc 17294  Atomscatm 39887  LHypclh 40608  LTrncltrn 40725  TEndoctendo 41376  DIsoCcdic 41796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-riota 7353  df-1st 7970  df-2nd 7971  df-dic 41797

This theorem is referenced by: (None)