Theorem cdleme51finvfvN 40660

Description: Part of proof of Lemma E in [Crawley] p. 113. TODO: fix comment. (Contributed by NM, 14-Apr-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdlemef50.b	⊢ 𝐵 = (Base‘𝐾)
cdlemef50.l	⊢ ≤ = (le‘𝐾)
cdlemef50.j	⊢ ∨ = (join‘𝐾)
cdlemef50.m	⊢ ∧ = (meet‘𝐾)
cdlemef50.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemef50.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemef50.u	⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
cdlemef50.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdlemefs50.e	⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
cdlemef50.f	⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
cdlemef51.v	⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
cdlemef51.n	⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
cdlemefs51.o	⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
cdlemef51.g	⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))

Assertion

Ref	Expression
cdleme51finvfvN	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (◡𝐹‘𝑋) = (𝐺‘𝑋))

Distinct variable groups:   𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧, ∧   ∨ ,𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   ≤ ,𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   𝐴,𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑐,𝑠,𝑣,𝑥,𝑦,𝑧   𝐸,𝑎,𝑏,𝑐,𝑥,𝑦,𝑧   𝐹,𝑎,𝑏,𝑐,𝑢,𝑣   𝐻,𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   𝐾,𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   𝑃,𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   𝑄,𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   𝑈,𝑎,𝑏,𝑐,𝑠,𝑡,𝑣,𝑥,𝑦,𝑧   𝑊,𝑎,𝑏,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   𝑋,𝑎,𝑐,𝑠,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧   𝐺,𝑠,𝑡,𝑥,𝑦,𝑧   𝑁,𝑎,𝑏,𝑐,𝑡,𝑢,𝑥,𝑦,𝑧   𝑂,𝑎,𝑏,𝑐,𝑥,𝑦,𝑧   𝑉,𝑎,𝑏,𝑐,𝑡,𝑢,𝑣,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐷(𝑢,𝑡)   𝑈(𝑢)   𝐸(𝑣,𝑢,𝑡,𝑠)   𝐹(𝑥,𝑦,𝑧,𝑡,𝑠)   𝐺(𝑣,𝑢,𝑎,𝑏,𝑐)   𝑁(𝑣,𝑠)   𝑂(𝑣,𝑢,𝑡,𝑠)   𝑉(𝑠)   𝑋(𝑏)

Proof of Theorem cdleme51finvfvN

Step	Hyp	Ref	Expression
1		cdlemef50.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemef50.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		cdlemef50.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		cdlemef50.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		cdlemef50.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemef50.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemef50.u	. . 3 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdlemef50.d	. . 3 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
9		cdlemefs50.e	. . 3 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
10		cdlemef50.f	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ if((𝑃 ≠ 𝑄 ∧ ¬ 𝑥 ≤ 𝑊), (℩𝑧 ∈ 𝐵 ∀𝑠 ∈ 𝐴 ((¬ 𝑠 ≤ 𝑊 ∧ (𝑠 ∨ (𝑥 ∧ 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 ≤ (𝑃 ∨ 𝑄), (℩𝑦 ∈ 𝐵 ∀𝑡 ∈ 𝐴 ((¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ (𝑃 ∨ 𝑄)) → 𝑦 = 𝐸)), ⦋𝑠 / 𝑡⦌𝐷) ∨ (𝑥 ∧ 𝑊)))), 𝑥))
11		cdlemef51.v	. . 3 ⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
12		cdlemef51.n	. . 3 ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
13		cdlemefs51.o	. . 3 ⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
14		cdlemef51.g	. . 3 ⊢ 𝐺 = (𝑎 ∈ 𝐵 ↦ if((𝑄 ≠ 𝑃 ∧ ¬ 𝑎 ≤ 𝑊), (℩𝑐 ∈ 𝐵 ∀𝑢 ∈ 𝐴 ((¬ 𝑢 ≤ 𝑊 ∧ (𝑢 ∨ (𝑎 ∧ 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 ≤ (𝑄 ∨ 𝑃), (℩𝑏 ∈ 𝐵 ∀𝑣 ∈ 𝐴 ((¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ (𝑄 ∨ 𝑃)) → 𝑏 = 𝑂)), ⦋𝑢 / 𝑣⦌𝑁) ∨ (𝑎 ∧ 𝑊)))), 𝑎))
15	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14	cdleme48fgv 40643	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝐹‘(𝐺‘𝑋)) = 𝑋)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme50f1o 40651	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹:𝐵–1-1-onto→𝐵)
17	16	adantr 480	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → 𝐹:𝐵–1-1-onto→𝐵)
18	1, 2, 3, 4, 5, 6, 11, 12, 13, 14	cdlemeg46fvcl 40611	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) ∈ 𝐵)
19		f1ocnvfv 7218	. . 3 ⊢ ((𝐹:𝐵–1-1-onto→𝐵 ∧ (𝐺‘𝑋) ∈ 𝐵) → ((𝐹‘(𝐺‘𝑋)) = 𝑋 → (◡𝐹‘𝑋) = (𝐺‘𝑋)))
20	17, 18, 19	syl2anc 584	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → ((𝐹‘(𝐺‘𝑋)) = 𝑋 → (◡𝐹‘𝑋) = (𝐺‘𝑋)))
21	15, 20	mpd 15	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑋 ∈ 𝐵) → (◡𝐹‘𝑋) = (𝐺‘𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ⦋csb 3845  ifcif 4474   class class class wbr 5093   ↦ cmpt 5174  ^◡ccnv 5618  –1-1-onto→wf1o 6486  ‘cfv 6487  ℩crio 7308  (class class class)co 7352  Basecbs 17126  lecple 17174  joincjn 18223  meetcmee 18224  Atomscatm 39368  HLchlt 39455  LHypclh 40089

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-riotaBAD 39058

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-undef 8209  df-proset 18206  df-poset 18225  df-plt 18240  df-lub 18256  df-glb 18257  df-join 18258  df-meet 18259  df-p0 18335  df-p1 18336  df-lat 18344  df-clat 18411  df-oposet 39281  df-ol 39283  df-oml 39284  df-covers 39371  df-ats 39372  df-atl 39403  df-cvlat 39427  df-hlat 39456  df-llines 39603  df-lplanes 39604  df-lvols 39605  df-lines 39606  df-psubsp 39608  df-pmap 39609  df-padd 39901  df-lhyp 40093

This theorem is referenced by:  cdleme51finvN  40661