Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pmod1i Structured version   Visualization version   GIF version

Theorem pmod1i 38314
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 2737 . . . . 5 (leβ€˜πΎ) = (leβ€˜πΎ)
2 eqid 2737 . . . . 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 38313 . . . 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 1119 . . 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 4189 . . . . 5 (π‘Œ ∩ 𝑍) βŠ† π‘Œ
9 simpll 766 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝐾 ∈ HL)
10 simplr2 1217 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ π‘Œ βŠ† 𝐴)
11 simplr1 1216 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑋 βŠ† 𝐴)
123, 5paddss2 38284 . . . . . 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 1372 . . . . 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 484 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ 𝐾 ∈ HL)
163, 4psubssat 38220 . . . . . . . 8 ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) β†’ 𝑍 βŠ† 𝐴)
17163ad2antr3 1191 . . . . . . 7 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ 𝑍 βŠ† 𝐴)
18 simpr2 1196 . . . . . . . 8 ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ π‘Œ βŠ† 𝐴)
19 ssinss1 4198 . . . . . . . 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 38283 . . . . . . 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 1372 . . . . . 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 408 . . . . 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 1218 . . . . . . . 8 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑍 ∈ 𝑆)
259, 24, 16syl2anc 585 . . . . . . 7 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ 𝑍 βŠ† 𝐴)
26 inss2 4190 . . . . . . . 8 (π‘Œ ∩ 𝑍) βŠ† 𝑍
273, 5paddss2 38284 . . . . . . . 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 1372 . . . . . 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 38307 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑍 ∈ 𝑆) β†’ (𝑍 + 𝑍) = 𝑍)
319, 24, 30syl2anc 585 . . . . . 6 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + 𝑍) = 𝑍)
3229, 31sseqtrd 3985 . . . . 5 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑍 + (π‘Œ ∩ 𝑍)) βŠ† 𝑍)
3323, 32sstrd 3955 . . . 4 (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) ∧ 𝑋 βŠ† 𝑍) β†’ (𝑋 + (π‘Œ ∩ 𝑍)) βŠ† 𝑍)
3414, 33ssind 4193 . . 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 3962 . 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 414 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ∩ cin 3910   βŠ† wss 3911  β€˜cfv 6497  (class class class)co 7358  lecple 17141  joincjn 18201  Atomscatm 37728  HLchlt 37815  PSubSpcpsubsp 37962  +𝑃cpadd 38261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-lat 18322  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-psubsp 37969  df-padd 38262
This theorem is referenced by:  pmod2iN  38315  pmodN  38316  pmodl42N  38317  hlmod1i  38322
  Copyright terms: Public domain W3C validator