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Theorem pmodl42N 35658
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 3948 . . . 4 ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))
2 simpl1 1228 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝐾 ∈ HL)
3 simpl3 1232 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑌𝑆)
4 eqid 2760 . . . . . . 7 (Atoms‘𝐾) = (Atoms‘𝐾)
5 pmodl42.s . . . . . . 7 𝑆 = (PSubSp‘𝐾)
64, 5psubssat 35561 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌𝑆) → 𝑌 ⊆ (Atoms‘𝐾))
72, 3, 6syl2anc 696 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑌 ⊆ (Atoms‘𝐾))
8 simpl2 1230 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋𝑆)
94, 5psubssat 35561 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑋𝑆) → 𝑋 ⊆ (Atoms‘𝐾))
102, 8, 9syl2anc 696 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋 ⊆ (Atoms‘𝐾))
11 simprl 811 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑍𝑆)
124, 5psubssat 35561 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍𝑆) → 𝑍 ⊆ (Atoms‘𝐾))
132, 11, 12syl2anc 696 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑍 ⊆ (Atoms‘𝐾))
14 pmodl42.p . . . . . . 7 + = (+𝑃𝐾)
154, 14paddssat 35621 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
162, 10, 13, 15syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + 𝑍) ⊆ (Atoms‘𝐾))
17 simprr 813 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑊𝑆)
185, 14paddclN 35649 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌𝑆𝑊𝑆) → (𝑌 + 𝑊) ∈ 𝑆)
192, 3, 17, 18syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + 𝑊) ∈ 𝑆)
204, 5psubssat 35561 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑊𝑆) → 𝑊 ⊆ (Atoms‘𝐾))
212, 17, 20syl2anc 696 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑊 ⊆ (Atoms‘𝐾))
224, 14sspadd1 35622 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑊 ⊆ (Atoms‘𝐾)) → 𝑌 ⊆ (𝑌 + 𝑊))
232, 7, 21, 22syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑌 ⊆ (𝑌 + 𝑊))
244, 5, 14pmod1i 35655 . . . . . 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 1110 . . . . 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 1490 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑌 + 𝑊)) = (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))
271, 26syl5reqr 2809 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍))))
2827oveq2d 6830 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
29 ssinss1 3984 . . . 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 35645 . . 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 1479 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))) = (𝑋 + (𝑌 + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))))
334, 14paddass 35645 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
342, 10, 7, 13, 33syl13anc 1479 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
354, 14padd12N 35646 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾))) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
362, 10, 7, 13, 35syl13anc 1479 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))
3734, 36eqtrd 2794 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
384, 14paddass 35645 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾) ∧ 𝑊 ⊆ (Atoms‘𝐾))) → ((𝑋 + 𝑌) + 𝑊) = (𝑋 + (𝑌 + 𝑊)))
392, 10, 7, 21, 38syl13anc 1479 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + 𝑌) + 𝑊) = (𝑋 + (𝑌 + 𝑊)))
4037, 39ineq12d 3958 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))))
41 incom 3948 . . . 4 ((𝑌 + (𝑋 + 𝑍)) ∩ (𝑋 + (𝑌 + 𝑊))) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍)))
4240, 41syl6eq 2810 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))))
434, 5psubssat 35561 . . . . 5 ((𝐾 ∈ HL ∧ (𝑌 + 𝑊) ∈ 𝑆) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
442, 19, 43syl2anc 696 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + 𝑊) ⊆ (Atoms‘𝐾))
455, 14paddclN 35649 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋𝑆𝑍𝑆) → (𝑋 + 𝑍) ∈ 𝑆)
462, 8, 11, 45syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + 𝑍) ∈ 𝑆)
475, 14paddclN 35649 . . . . 5 ((𝐾 ∈ HL ∧ 𝑌𝑆 ∧ (𝑋 + 𝑍) ∈ 𝑆) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
482, 3, 46, 47syl3anc 1477 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑌 + (𝑋 + 𝑍)) ∈ 𝑆)
494, 14sspadd1 35622 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾) ∧ 𝑍 ⊆ (Atoms‘𝐾)) → 𝑋 ⊆ (𝑋 + 𝑍))
502, 10, 13, 49syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋 ⊆ (𝑋 + 𝑍))
514, 14sspadd2 35623 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 + 𝑍) ⊆ (Atoms‘𝐾) ∧ 𝑌 ⊆ (Atoms‘𝐾)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
522, 16, 7, 51syl3anc 1477 . . . . 5 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (𝑋 + 𝑍) ⊆ (𝑌 + (𝑋 + 𝑍)))
5350, 52sstrd 3754 . . . 4 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → 𝑋 ⊆ (𝑌 + (𝑋 + 𝑍)))
544, 5, 14pmod1i 35655 . . . . 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 1110 . . . 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 1490 . . 3 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → ((𝑋 + (𝑌 + 𝑊)) ∩ (𝑌 + (𝑋 + 𝑍))) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
5742, 56eqtrd 2794 . 2 (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = (𝑋 + ((𝑌 + 𝑊) ∩ (𝑌 + (𝑋 + 𝑍)))))
5828, 32, 573eqtr4rd 2805 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 1072   = wceq 1632  wcel 2139  cin 3714  wss 3715  cfv 6049  (class class class)co 6814  Atomscatm 35071  HLchlt 35158  PSubSpcpsubsp 35303  +𝑃cpadd 35602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-preset 17149  df-poset 17167  df-plt 17179  df-lub 17195  df-glb 17196  df-join 17197  df-meet 17198  df-p0 17260  df-lat 17267  df-clat 17329  df-oposet 34984  df-ol 34986  df-oml 34987  df-covers 35074  df-ats 35075  df-atl 35106  df-cvlat 35130  df-hlat 35159  df-psubsp 35310  df-padd 35603
This theorem is referenced by:  pl42lem4N  35789
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