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Theorem pmod1i 39230
Description: The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)
Hypotheses
Ref Expression
pmod.a 𝐴 = (Atomsβ€˜πΎ)
pmod.s 𝑆 = (PSubSpβ€˜πΎ)
pmod.p + = (+π‘ƒβ€˜πΎ)
Assertion
Ref Expression
pmod1i ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ (𝑋 βŠ† 𝑍 β†’ ((𝑋 + π‘Œ) ∩ 𝑍) = (𝑋 + (π‘Œ ∩ 𝑍))))
Proof of Theorem pmod1i
StepHypRef Expression
1 eqid 2726 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2726 . . . . 5 (joinβ€˜πΎ) = (joinβ€˜πΎ)
3 pmod.a . . . . 5 𝐴 = (Atomsβ€˜πΎ)
4 pmod.s . . . . 5 𝑆 = (PSubSpβ€˜πΎ)
5 pmod.p . . . . 5 + = (+π‘ƒβ€˜πΎ)
61, 2, 3, 4, 5pmodlem2 39229 . . . 4 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 βŠ† 𝑍) β†’ ((𝑋 + π‘Œ) ∩ 𝑍) βŠ† (𝑋 + (π‘Œ ∩ 𝑍)))
763expa 1115 . . 3 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ ((𝑋 + π‘Œ) ∩ 𝑍) βŠ† (𝑋 + (π‘Œ ∩ 𝑍)))
8 inss1 4223 . . . . 5 (π‘Œ ∩ 𝑍) βŠ† π‘Œ
9 simpll 764 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝐾 ∈ HL)
10 simplr2 1213 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ π‘Œ βŠ† 𝐴)
11 simplr1 1212 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑋 βŠ† 𝐴)
123, 5paddss2 39200 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† 𝐴) β†’ ((π‘Œ ∩ 𝑍) βŠ† π‘Œ β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑋 + π‘Œ)))
139, 10, 11, 12syl3anc 1368 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ ((π‘Œ ∩ 𝑍) βŠ† π‘Œ β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑋 + π‘Œ)))
148, 13mpi 20 . . . 4 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑋 + π‘Œ))
15 simpl 482 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ 𝐾 ∈ HL)
163, 4psubssat 39136 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) β†’ 𝑍 βŠ† 𝐴)
17163ad2antr3 1187 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ 𝑍 βŠ† 𝐴)
18 simpr2 1192 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ π‘Œ βŠ† 𝐴)
19 ssinss1 4232 . . . . . . . 8 (π‘Œ βŠ† 𝐴 β†’ (π‘Œ ∩ 𝑍) βŠ† 𝐴)
2018, 19syl 17 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ (π‘Œ ∩ 𝑍) βŠ† 𝐴)
213, 5paddss1 39199 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ (π‘Œ ∩ 𝑍) βŠ† 𝐴) β†’ (𝑋 βŠ† 𝑍 β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + (π‘Œ ∩ 𝑍))))
2215, 17, 20, 21syl3anc 1368 . . . . . 6 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ (𝑋 βŠ† 𝑍 β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + (π‘Œ ∩ 𝑍))))
2322imp 406 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + (π‘Œ ∩ 𝑍)))
24 simplr3 1214 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑍 ∈ 𝑆)
259, 24, 16syl2anc 583 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑍 βŠ† 𝐴)
26 inss2 4224 . . . . . . . 8 (π‘Œ ∩ 𝑍) βŠ† 𝑍
273, 5paddss2 39200 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ ((π‘Œ ∩ 𝑍) βŠ† 𝑍 β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + 𝑍)))
2826, 27mpi 20 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍 βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + 𝑍))
299, 25, 25, 28syl3anc 1368 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† (𝑍 + 𝑍))
304, 5paddidm 39223 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) β†’ (𝑍 + 𝑍) = 𝑍)
319, 24, 30syl2anc 583 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + 𝑍) = 𝑍)
3229, 31sseqtrd 4017 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† 𝑍)
3323, 32sstrd 3987 . . . 4 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† 𝑍)
3414, 33ssind 4227 . . 3 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† ((𝑋 + π‘Œ) ∩ 𝑍))
357, 34eqssd 3994 . 2 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ ((𝑋 + π‘Œ) ∩ 𝑍) = (𝑋 + (π‘Œ ∩ 𝑍)))
3635ex 412 1 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ (𝑋 βŠ† 𝑍 β†’ ((𝑋 + π‘Œ) ∩ 𝑍) = (𝑋 + (π‘Œ ∩ 𝑍))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3942   βŠ† wss 3943  β€˜cfv 6536  (class class class)co 7404  lecple 17211  joincjn 18274  Atomscatm 38644  HLchlt 38731  PSubSpcpsubsp 38878  +𝑃cpadd 39177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-lat 18395  df-covers 38647  df-ats 38648  df-atl 38679  df-cvlat 38703  df-hlat 38732  df-psubsp 38885  df-padd 39178
This theorem is referenced by:  pmod2iN  39231  pmodN  39232  pmodl42N  39233  hlmod1i  39238
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