Theorem cdleme51finvN 40990

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
cdleme51finvN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ◡𝐹 = 𝐺)

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   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)

Proof of Theorem cdleme51finvN

Dummy variable 𝑒 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cdlemef50.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		cdlemef50.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
3		cdlemef50.j	. . . . 5 ⊢ ∨ = (join‘𝐾)
4		cdlemef50.m	. . . . 5 ⊢ ∧ = (meet‘𝐾)
5		cdlemef50.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
6		cdlemef50.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
7		cdlemef50.u	. . . . 5 ⊢ 𝑈 = ((𝑃 ∨ 𝑄) ∧ 𝑊)
8		cdlemef50.d	. . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
9		cdlemefs50.e	. . . . 5 ⊢ 𝐸 = ((𝑃 ∨ 𝑄) ∧ (𝐷 ∨ ((𝑠 ∨ 𝑡) ∧ 𝑊)))
10		cdlemef50.f	. . . . 5 ⊢ 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	1, 2, 3, 4, 5, 6, 7, 8, 9, 10	cdleme50f1o 40980	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹:𝐵–1-1-onto→𝐵)
12		dff1o4 6777	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐵))
13	11, 12	sylib 218	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐹 Fn 𝐵 ∧ ◡𝐹 Fn 𝐵))
14	13	simprd 495	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ◡𝐹 Fn 𝐵)
15		cdlemef51.v	. . . . 5 ⊢ 𝑉 = ((𝑄 ∨ 𝑃) ∧ 𝑊)
16		cdlemef51.n	. . . . 5 ⊢ 𝑁 = ((𝑣 ∨ 𝑉) ∧ (𝑃 ∨ ((𝑄 ∨ 𝑣) ∧ 𝑊)))
17		cdlemefs51.o	. . . . 5 ⊢ 𝑂 = ((𝑄 ∨ 𝑃) ∧ (𝑁 ∨ ((𝑢 ∨ 𝑣) ∧ 𝑊)))
18		cdlemef51.g	. . . . 5 ⊢ 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))
19	1, 2, 3, 4, 5, 6, 15, 16, 17, 18	cdleme50f1o 40980	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐺:𝐵–1-1-onto→𝐵)
20	19	3com23 1127	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺:𝐵–1-1-onto→𝐵)
21		f1ofn 6770	. . 3 ⊢ (𝐺:𝐵–1-1-onto→𝐵 → 𝐺 Fn 𝐵)
22	20, 21	syl 17	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐺 Fn 𝐵)
23	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 16, 17, 18	cdleme51finvfvN 40989	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑒 ∈ 𝐵) → (◡𝐹‘𝑒) = (𝐺‘𝑒))
24	14, 22, 23	eqfnfvd 6975	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ◡𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2930  ∀wral 3049  ⦋csb 3833  ifcif 4456   class class class wbr 5074   ↦ cmpt 5155  ^◡ccnv 5619   Fn wfn 6482  –1-1-onto→wf1o 6486  ‘cfv 6487  ℩crio 7312  (class class class)co 7356  Basecbs 17168  lecple 17216  joincjn 18266  meetcmee 18267  Atomscatm 39697  HLchlt 39784  LHypclh 40418

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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678  ax-riotaBAD 39387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-rmo 3340  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-iin 4926  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  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 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-undef 8212  df-proset 18249  df-poset 18268  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18387  df-clat 18454  df-oposet 39610  df-ol 39612  df-oml 39613  df-covers 39700  df-ats 39701  df-atl 39732  df-cvlat 39756  df-hlat 39785  df-llines 39932  df-lplanes 39933  df-lvols 39934  df-lines 39935  df-psubsp 39937  df-pmap 39938  df-padd 40230  df-lhyp 40422

This theorem is referenced by:  cdleme51finvtrN  40992