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Theorem omlfh3N 34053
Description: Foulis-Holland Theorem, part 3. Dual of omlfh1N 34052. (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 2621 . . . . . . 7 (oc‘𝐾) = (oc‘𝐾)
3 omlfh1.c . . . . . . 7 𝐶 = (cm‘𝐾)
41, 2, 3cmt4N 34046 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑌𝐵) → (𝑋𝐶𝑌 ↔ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌)))
543adant3r3 1273 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑌 ↔ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌)))
61, 2, 3cmt4N 34046 . . . . . 6 ((𝐾 ∈ OML ∧ 𝑋𝐵𝑍𝐵) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍)))
763adant3r2 1272 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋𝐶𝑍 ↔ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍)))
85, 7anbi12d 746 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋𝐶𝑌𝑋𝐶𝑍) ↔ (((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑌) ∧ ((oc‘𝐾)‘𝑋)𝐶((oc‘𝐾)‘𝑍))))
9 simpl 473 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OML)
10 omlop 34035 . . . . . . . 8 (𝐾 ∈ OML → 𝐾 ∈ OP)
1110adantr 481 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OP)
12 simpr1 1065 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑋𝐵)
131, 2opoccl 33988 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑋𝐵) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
1411, 12, 13syl2anc 692 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑋) ∈ 𝐵)
15 simpr2 1066 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑌𝐵)
161, 2opoccl 33988 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑌𝐵) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
1711, 15, 16syl2anc 692 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑌) ∈ 𝐵)
18 simpr3 1067 . . . . . . 7 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝑍𝐵)
191, 2opoccl 33988 . . . . . . 7 ((𝐾 ∈ OP ∧ 𝑍𝐵) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
2011, 18, 19syl2anc 692 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘𝑍) ∈ 𝐵)
2114, 17, 203jca 1240 . . . . 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 34052 . . . . . . 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 6157 . . . . . 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 1261 . . . . 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 230 . . 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 1258 . 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 34034 . . . . . 6 (𝐾 ∈ OML → 𝐾 ∈ OL)
3130adantr 481 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ OL)
32 omllat 34036 . . . . . . 7 (𝐾 ∈ OML → 𝐾 ∈ Lat)
3332adantr 481 . . . . . 6 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → 𝐾 ∈ Lat)
341, 22latjcl 16979 . . . . . 6 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
3533, 17, 20, 34syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
361, 22, 23, 2oldmm2 34012 . . . . 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 1323 . . . 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 34018 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑌𝐵𝑍𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = (𝑌 𝑍))
3931, 15, 18, 38syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍))) = (𝑌 𝑍))
4039oveq2d 6626 . . . 4 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 ((oc‘𝐾)‘(((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))) = (𝑋 (𝑌 𝑍)))
4137, 40eqtr2d 2656 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (𝑋 (𝑌 𝑍)) = ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
42413adant3 1079 . 2 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) (((oc‘𝐾)‘𝑌) ((oc‘𝐾)‘𝑍)))))
431, 23latmcl 16980 . . . . . 6 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑌) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
4433, 14, 17, 43syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) ∈ 𝐵)
451, 23latmcl 16980 . . . . . 6 ((𝐾 ∈ Lat ∧ ((oc‘𝐾)‘𝑋) ∈ 𝐵 ∧ ((oc‘𝐾)‘𝑍) ∈ 𝐵) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
4633, 14, 20, 45syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)) ∈ 𝐵)
471, 22, 23, 2oldmj1 34015 . . . . 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 1323 . . . 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 34014 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑌𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 𝑌))
5031, 12, 15, 49syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌))) = (𝑋 𝑌))
511, 22, 23, 2oldmm4 34014 . . . . . 6 ((𝐾 ∈ OL ∧ 𝑋𝐵𝑍𝐵) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 𝑍))
5231, 12, 18, 51syl3anc 1323 . . . . 5 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((oc‘𝐾)‘(((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍))) = (𝑋 𝑍))
5350, 52oveq12d 6628 . . . 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 2656 . . 3 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑌) (𝑋 𝑍)) = ((oc‘𝐾)‘((((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑌)) (((oc‘𝐾)‘𝑋) ((oc‘𝐾)‘𝑍)))))
55543adant3 1079 . 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 2665 1 ((𝐾 ∈ OML ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋𝐶𝑌𝑋𝐶𝑍)) → (𝑋 (𝑌 𝑍)) = ((𝑋 𝑌) (𝑋 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987   class class class wbr 4618  cfv 5852  (class class class)co 6610  Basecbs 15788  occoc 15877  joincjn 16872  meetcmee 16873  Latclat 16973  OPcops 33966  cmccmtN 33967  OLcol 33968  OMLcoml 33969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-preset 16856  df-poset 16874  df-lub 16902  df-glb 16903  df-join 16904  df-meet 16905  df-p0 16967  df-lat 16974  df-oposet 33970  df-cmtN 33971  df-ol 33972  df-oml 33973
This theorem is referenced by: (None)
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