Theorem occon 10793

Description: Contraposition law for orthogonal complement.

Assertion

Ref	Expression
occon	|- ((A C_ ~H /\ B C_ ~H) -> (A C_ B -> (_|_` B) C_ (_|_` A)))

Proof of Theorem occon

Step	Hyp	Ref	Expression
1		ssel 2615	. . . . . . . . . 10 |- (A C_ B -> (y e. A -> y e. B))
2	1	imim1d 33	. . . . . . . . 9 |- (A C_ B -> ((y e. B -> (x .ih y) = 0) -> (y e. A -> (x .ih y) = 0)))
3	2	alimdv 1668	. . . . . . . 8 |- (A C_ B -> (A.y(y e. B -> (x .ih y) = 0) -> A.y(y e. A -> (x .ih y) = 0)))
4		df-ral 2109	. . . . . . . 8 |- (A.y e. B (x .ih y) = 0 <-> A.y(y e. B -> (x .ih y) = 0))
5		df-ral 2109	. . . . . . . 8 |- (A.y e. A (x .ih y) = 0 <-> A.y(y e. A -> (x .ih y) = 0))
6	3, 4, 5	3imtr4g 612	. . . . . . 7 |- (A C_ B -> (A.y e. B (x .ih y) = 0 -> A.y e. A (x .ih y) = 0))
7	6	a1d 15	. . . . . 6 |- (A C_ B -> (x e. ~H -> (A.y e. B (x .ih y) = 0 -> A.y e. A (x .ih y) = 0)))
8	7	r19.21aiv 2175	. . . . 5 |- (A C_ B -> A.x e. ~H (A.y e. B (x .ih y) = 0 -> A.y e. A (x .ih y) = 0))
9		ss2rab 2683	. . . . 5 |- ({x e. ~H | A.y e. B (x .ih y) = 0} C_ {x e. ~H | A.y e. A (x .ih y) = 0} <-> A.x e. ~H (A.y e. B (x .ih y) = 0 -> A.y e. A (x .ih y) = 0))
10	8, 9	sylibr 217	. . . 4 |- (A C_ B -> {x e. ~H | A.y e. B (x .ih y) = 0} C_ {x e. ~H | A.y e. A (x .ih y) = 0})
11	10	adantl 424	. . 3 |- (((A C_ ~H /\ B C_ ~H) /\ A C_ B) -> {x e. ~H | A.y e. B (x .ih y) = 0} C_ {x e. ~H | A.y e. A (x .ih y) = 0})
12		ocval 10786	. . . 4 |- (B C_ ~H -> (_|_` B) = {x e. ~H | A.y e. B (x .ih y) = 0})
13	12	ad2antlr 441	. . 3 |- (((A C_ ~H /\ B C_ ~H) /\ A C_ B) -> (_|_` B) = {x e. ~H | A.y e. B (x .ih y) = 0})
14		ocval 10786	. . . 4 |- (A C_ ~H -> (_|_` A) = {x e. ~H | A.y e. A (x .ih y) = 0})
15	14	ad2antrr 440	. . 3 |- (((A C_ ~H /\ B C_ ~H) /\ A C_ B) -> (_|_` A) = {x e. ~H | A.y e. A (x .ih y) = 0})
16	11, 13, 15	3sstr4d 2660	. 2 |- (((A C_ ~H /\ B C_ ~H) /\ A C_ B) -> (_|_` B) C_ (_|_` A))
17	16	ex 402	1 |- ((A C_ ~H /\ B C_ ~H) -> (A C_ B -> (_|_` B) C_ (_|_` A)))

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108   C_ wss 2593  ` cfv 3998  (class class class)co 4884  0cc0 6386  ~Hchil 10420   .ih csp 10425  _|_cort 10431

This theorem is referenced by:  occon2 10794  ococin 10930  chsscon3i 11017  shjshsi 11048

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-hilex 10501

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-oc 10757

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