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Theorem pmodl42N 35455
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 3838 . . . 4 ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))
2 simpl1 1084 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝐾 ∈ HL)
3 simpl3 1086 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑌𝑆)
4 eqid 2651 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pmodl42.s . . . . . . 7 𝑆 = (PSubSp‘𝐾)
64, 5psubssat 35358 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
72, 3, 6syl2anc 694 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑌 ⊆ (Atoms‘𝐾))
8 simpl2 1085 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋𝑆)
94, 5psubssat 35358 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
102, 8, 9syl2anc 694 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋 ⊆ (Atoms‘𝐾))
11 simprl 809 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑍𝑆)
124, 5psubssat 35358 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍𝑆) → 𝑍 ⊆ (Atoms‘𝐾))
132, 11, 12syl2anc 694 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑍 ⊆ (Atoms‘𝐾))
14 pmodl42.p . . . . . . 7 + = (+𝑃𝐾)
154, 14paddssat 35418 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
162, 10, 13, 15syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
17 simprr 811 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑊𝑆)
185, 14paddclN 35446 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌𝑆𝑊𝑆) → (𝑌 + 𝑊) ∈ 𝑆)
192, 3, 17, 18syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + 𝑊) ∈ 𝑆)
204, 5psubssat 35358 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝑆) → 𝑊 ⊆ (Atoms‘𝐾))
212, 17, 20syl2anc 694 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑊 ⊆ (Atoms‘𝐾))
224, 14sspadd1 35419 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑊 ⊆ (Atoms‘𝐾)) → 𝑌 ⊆ (𝑌 + 𝑊))
232, 7, 21, 22syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑌 ⊆ (𝑌 + 𝑊))
244, 5, 14pmod1i 35452 . . . . . 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 1280 . . . . 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 1379 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))
271, 26syl5reqr 2700 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍))))
2827oveq2d 6706 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
29 ssinss1 3874 . . . 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 35442 . . 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 1368 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))))
334, 14paddass 35442 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
342, 10, 7, 13, 33syl13anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
354, 14padd12N 35443 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾))) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
362, 10, 7, 13, 35syl13anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
3734, 36eqtrd 2685 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
384, 14paddass 35442 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑊 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + 𝑊) = (𝑋 + (𝑌 + 𝑊)))
392, 10, 7, 21, 38syl13anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + 𝑊) = (𝑋 + (𝑌 + 𝑊)))
4037, 39ineq12d 3848 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))))
41 incom 3838 . . . 4 ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍)))
4240, 41syl6eq 2701 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))))
434, 5psubssat 35358 . . . . 5 ((𝐾 ∈ HL ∧ (𝑌 + 𝑊) ∈ 𝑆) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
442, 19, 43syl2anc 694 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
455, 14paddclN 35446 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑍𝑆) → (𝑋 + 𝑍) ∈ 𝑆)
462, 8, 11, 45syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + 𝑍) ∈ 𝑆)
475, 14paddclN 35446 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑆 ∧ (𝑋 + 𝑍) ∈ 𝑆) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
482, 3, 46, 47syl3anc 1366 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
494, 14sspadd1 35419 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑍))
502, 10, 13, 49syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋 ⊆ (𝑋 + 𝑍))
514, 14sspadd2 35420 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 + 𝑍) ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
522, 16, 7, 51syl3anc 1366 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
5350, 52sstrd 3646 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋 ⊆ (𝑌 + (𝑋 + 𝑍)))
544, 5, 14pmod1i 35452 . . . . 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 1280 . . . 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 1379 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
5742, 56eqtrd 2685 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
5828, 32, 573eqtr4rd 2696 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 383  w3a 1054   = wceq 1523  wcel 2030  cin 3606  wss 3607  cfv 5926  (class class class)co 6690  Atomscatm 34868  HLchlt 34955  PSubSpcpsubsp 35100  +𝑃cpadd 35399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-lat 17093  df-clat 17155  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-psubsp 35107  df-padd 35400
This theorem is referenced by:  pl42lem4N  35586
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