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Theorem pmodl42N 36867
Description: Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmodl42.s 𝑆 = (PSubSp‘𝐾)
pmodl42.p + = (+𝑃𝐾)
Assertion
Ref Expression
pmodl42N (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))

Proof of Theorem pmodl42N
StepHypRef Expression
1 incom 4175 . . . 4 ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))
2 simpl1 1183 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝐾 ∈ HL)
3 simpl3 1185 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑌𝑆)
4 eqid 2818 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pmodl42.s . . . . . . 7 𝑆 = (PSubSp‘𝐾)
64, 5psubssat 36770 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
72, 3, 6syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑌 ⊆ (Atoms‘𝐾))
8 simpl2 1184 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋𝑆)
94, 5psubssat 36770 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
102, 8, 9syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋 ⊆ (Atoms‘𝐾))
11 simprl 767 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑍𝑆)
124, 5psubssat 36770 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍𝑆) → 𝑍 ⊆ (Atoms‘𝐾))
132, 11, 12syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑍 ⊆ (Atoms‘𝐾))
14 pmodl42.p . . . . . . 7 + = (+𝑃𝐾)
154, 14paddssat 36830 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
162, 10, 13, 15syl3anc 1363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
17 simprr 769 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑊𝑆)
185, 14paddclN 36858 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌𝑆𝑊𝑆) → (𝑌 + 𝑊) ∈ 𝑆)
192, 3, 17, 18syl3anc 1363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + 𝑊) ∈ 𝑆)
204, 5psubssat 36770 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝑆) → 𝑊 ⊆ (Atoms‘𝐾))
212, 17, 20syl2anc 584 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑊 ⊆ (Atoms‘𝐾))
224, 14sspadd1 36831 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑊 ⊆ (Atoms‘𝐾)) → 𝑌 ⊆ (𝑌 + 𝑊))
232, 7, 21, 22syl3anc 1363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑌 ⊆ (𝑌 + 𝑊))
244, 5, 14pmod1i 36864 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑌 ⊆ (Atoms‘𝐾) ∧ (𝑋 + 𝑍) ⊆ (Atoms‘𝐾) ∧ (𝑌 + 𝑊) ∈ 𝑆)) → (𝑌 ⊆ (𝑌 + 𝑊) → ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))))
25243impia 1109 . . . . 5 ((𝐾 ∈ HL ∧ (𝑌 ⊆ (Atoms‘𝐾) ∧ (𝑋 + 𝑍) ⊆ (Atoms‘𝐾) ∧ (𝑌 + 𝑊) ∈ 𝑆) ∧ 𝑌 ⊆ (𝑌 + 𝑊)) → ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))
262, 7, 16, 19, 23, 25syl131anc 1375 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))
271, 26syl5reqr 2868 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍))))
2827oveq2d 7161 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
29 ssinss1 4211 . . . 4 ((𝑋 + 𝑍) ⊆ (Atoms‘𝐾) → ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)) ⊆ (Atoms‘𝐾))
3016, 29syl 17 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)) ⊆ (Atoms‘𝐾))
314, 14paddass 36854 . . 3 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)) ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))))
322, 10, 7, 30, 31syl13anc 1364 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))))
334, 14paddass 36854 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
342, 10, 7, 13, 33syl13anc 1364 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
354, 14padd12N 36855 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾))) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
362, 10, 7, 13, 35syl13anc 1364 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
3734, 36eqtrd 2853 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
384, 14paddass 36854 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑊 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + 𝑊) = (𝑋 + (𝑌 + 𝑊)))
392, 10, 7, 21, 38syl13anc 1364 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + 𝑊) = (𝑋 + (𝑌 + 𝑊)))
4037, 39ineq12d 4187 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))))
41 incom 4175 . . . 4 ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍)))
4240, 41syl6eq 2869 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))))
434, 5psubssat 36770 . . . . 5 ((𝐾 ∈ HL ∧ (𝑌 + 𝑊) ∈ 𝑆) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
442, 19, 43syl2anc 584 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
455, 14paddclN 36858 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑍𝑆) → (𝑋 + 𝑍) ∈ 𝑆)
462, 8, 11, 45syl3anc 1363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + 𝑍) ∈ 𝑆)
475, 14paddclN 36858 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑆 ∧ (𝑋 + 𝑍) ∈ 𝑆) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
482, 3, 46, 47syl3anc 1363 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
494, 14sspadd1 36831 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑍))
502, 10, 13, 49syl3anc 1363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋 ⊆ (𝑋 + 𝑍))
514, 14sspadd2 36832 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 + 𝑍) ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
522, 16, 7, 51syl3anc 1363 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
5350, 52sstrd 3974 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋 ⊆ (𝑌 + (𝑋 + 𝑍)))
544, 5, 14pmod1i 36864 . . . . 5 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ (𝑌 + 𝑊) ⊆ (Atoms‘𝐾) ∧ (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)) → (𝑋 ⊆ (𝑌 + (𝑋 + 𝑍)) → ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍))))))
55543impia 1109 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ (𝑌 + 𝑊) ⊆ (Atoms‘𝐾) ∧ (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆) ∧ 𝑋 ⊆ (𝑌 + (𝑋 + 𝑍))) → ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
562, 10, 44, 48, 53, 55syl131anc 1375 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
5742, 56eqtrd 2853 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
5828, 32, 573eqtr4rd 2864 1 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  cin 3932  wss 3933  cfv 6348  (class class class)co 7145  Atomscatm 36279  HLchlt 36366  PSubSpcpsubsp 36512  +𝑃cpadd 36811
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-mpo 7150  df-1st 7678  df-2nd 7679  df-proset 17526  df-poset 17544  df-plt 17556  df-lub 17572  df-glb 17573  df-join 17574  df-meet 17575  df-p0 17637  df-lat 17644  df-clat 17706  df-oposet 36192  df-ol 36194  df-oml 36195  df-covers 36282  df-ats 36283  df-atl 36314  df-cvlat 36338  df-hlat 36367  df-psubsp 36519  df-padd 36812
This theorem is referenced by:  pl42lem4N  36998
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