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Theorem omlfh3N 36275
Description: Foulis-Holland Theorem, part 3. Dual of omlfh1N 36274. (Contributed by NM, 8-Nov-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
omlfh1.b 𝐵 = (Base‘𝐾)
omlfh1.j = (join‘𝐾)
omlfh1.m = (meet‘𝐾)
omlfh1.c 𝐶 = (cm‘𝐾)
Assertion
Ref Expression
omlfh3N ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))

Proof of Theorem omlfh3N
StepHypRef Expression
1 omlfh1.b . . . . . . 7 𝐵 = (Base‘𝐾)
2 eqid 2818 . . . . . . 7 (oc‘𝐾) = (oc‘𝐾)
3 omlfh1.c . . . . . . 7 𝐶 = (cm‘𝐾)
41, 2, 3cmt4N 36268 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌)))
543adant3r3 1176 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 ↔ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌)))
61, 2, 3cmt4N 36268 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍)))
763adant3r2 1175 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍)))
85, 7anbi12d 630 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑋𝐶𝑍) ↔ (((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌) ∧ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍))))
9 simpl 483 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OML)
10 omlop 36257 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OP)
1110adantr 481 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OP)
12 simpr1 1186 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
131, 2opoccl 36210 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
1411, 12, 13syl2anc 584 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
15 simpr2 1187 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
161, 2opoccl 36210 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
1711, 15, 16syl2anc 584 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
18 simpr3 1188 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
191, 2opoccl 36210 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
2011, 18, 19syl2anc 584 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
2114, 17, 203jca 1120 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵))
22 omlfh1.j . . . . . . . 8 = (join‘𝐾)
23 omlfh1.m . . . . . . . 8 = (meet‘𝐾)
241, 22, 23, 3omlfh1N 36274 . . . . . . 7 ((𝐾 ∈ OML ∧ (((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) ∧ (((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌) ∧ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍))) → (((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = ((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))
2524fveq2d 6667 . . . . . 6 ((𝐾 ∈ OML ∧ (((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) ∧ (((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌) ∧ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍))) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
26253exp 1111 . . . . 5 (𝐾 ∈ OML → ((((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → ((((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌) ∧ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))))
279, 21, 26sylc 65 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌) ∧ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))))
288, 27sylbid 241 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑋𝐶𝑍) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))))))
29283impia 1109 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
30 omlol 36256 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ OL)
3130adantr 481 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OL)
32 omllat 36258 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ Lat)
3332adantr 481 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
341, 22latjcl 17649 . . . . . 6 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
3533, 17, 20, 34syl3anc 1363 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
361, 22, 23, 2oldmm2 36234 . . . . 5 ((𝐾 ∈ OL ∧ 𝑋𝐵 ∧ (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
3731, 12, 35, 36syl3anc 1363 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
381, 22, 23, 2oldmj4 36240 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑍𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = (𝑌 𝑍))
3931, 15, 18, 38syl3anc 1363 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = (𝑌 𝑍))
4039oveq2d 7161 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 (𝑌 𝑍)))
4137, 40eqtr2d 2854 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
42413adant3 1124 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
431, 23latmcl 17650 . . . . . 6 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
4433, 14, 17, 43syl3anc 1363 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
451, 23latmcl 17650 . . . . . 6 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
4633, 14, 20, 45syl3anc 1363 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
471, 22, 23, 2oldmj1 36237 . . . . 5 ((𝐾 ∈ OL ∧ (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵 ∧ (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵) → ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
4831, 44, 46, 47syl3anc 1363 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
491, 22, 23, 2oldmm4 36236 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 𝑌))
5031, 12, 15, 49syl3anc 1363 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 𝑌))
511, 22, 23, 2oldmm4 36236 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑍𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 𝑍))
5231, 12, 18, 51syl3anc 1363 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 𝑍))
5350, 52oveq12d 7163 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))) = ((𝑋 𝑌) (𝑋 𝑍)))
5448, 53eqtr2d 2854 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) = ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
55543adant3 1124 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → ((𝑋 𝑌) (𝑋 𝑍)) = ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
5629, 42, 553eqtr4d 2863 1 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  Basecbs 16471  occoc 16561  joincjn 17542  meetcmee 17543  Latclat 17643  OPcops 36188  cmccmtN 36189  OLcol 36190  OMLcoml 36191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-proset 17526  df-poset 17544  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-oposet 36192  df-cmtN 36193  df-ol 36194  df-oml 36195
This theorem is referenced by: (None)
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