Theorem occont 9155

Description: Contraposition law for orthogonal complement.

Assertion

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

Proof of Theorem occont

Step	Hyp	Ref	Expression
1		ssel 2066	. . . . . . . . . 10 |- (A (_ B -> (y e. A -> y e. B))
2	1	imim1d 28	. . . . . . . . 9 |- (A (_ B -> ((y e. B -> (x .ih y) = 0) -> (y e. A -> (x .ih y) = 0)))
3	2	19.20dv 1291	. . . . . . . 8 |- (A (_ B -> (A.y(y e. B -> (x .ih y) = 0) -> A.y(y e. A -> (x .ih y) = 0)))
4		df-ral 1652	. . . . . . . 8 |- (A.y e. B (x .ih y) = 0 <-> A.y(y e. B -> (x .ih y) = 0))
5		df-ral 1652	. . . . . . . 8 |- (A.y e. A (x .ih y) = 0 <-> A.y(y e. A -> (x .ih y) = 0))
6	3, 4, 5	3imtr4g 555	. . . . . . 7 |- (A (_ B -> (A.y e. B (x .ih y) = 0 -> A.y e. A (x .ih y) = 0))
7	6	a1d 12	. . . . . 6 |- (A (_ B -> (x e. H~ -> (A.y e. B (x .ih y) = 0 -> A.y e. A (x .ih y) = 0)))
8	7	r19.21aiv 1716	. . . . 5 |- (A (_ B -> A.x e. H~ (A.y e. B (x .ih y) = 0 -> A.y e. A (x .ih y) = 0))
9		ss2rab 2126	. . . . 5 |- ({x e. H~ | A.y e. B (x .ih y) = 0} (_ {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 200	. . . 4 |- (A (_ B -> {x e. H~ | A.y e. B (x .ih y) = 0} (_ {x e. H~ | A.y e. A (x .ih y) = 0})
11	10	adantl 390	. . 3 |- (((A (_ H~ /\ B (_ H~) /\ A (_ B) -> {x e. H~ | A.y e. B (x .ih y) = 0} (_ {x e. H~ | A.y e. A (x .ih y) = 0})
12		ocvalt 9148	. . . 4 |- (B (_ H~ -> (_|_` B) = {x e. H~ | A.y e. B (x .ih y) = 0})
13	12	ad2antlr 407	. . 3 |- (((A (_ H~ /\ B (_ H~) /\ A (_ B) -> (_|_` B) = {x e. H~ | A.y e. B (x .ih y) = 0})
14		ocvalt 9148	. . . 4 |- (A (_ H~ -> (_|_` A) = {x e. H~ | A.y e. A (x .ih y) = 0})
15	14	ad2antrr 406	. . 3 |- (((A (_ H~ /\ B (_ H~) /\ A (_ B) -> (_|_` A) = {x e. H~ | A.y e. A (x .ih y) = 0})
16	11, 13, 15	3sstr4d 2107	. 2 |- (((A (_ H~ /\ B (_ H~) /\ A (_ B) -> (_|_` B) (_ (_|_` A))
17	16	ex 373	1 |- ((A (_ H~ /\ B (_ H~) -> (A (_ B -> (_|_` B) (_ (_|_` A)))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  A.wral 1648  {crab 1651   (_ wss 2050  ` cfv 3188  (class class class)co 3969  0cc0 5246  H~chil 8783   .ih csp 8788  _|_cort 8794

This theorem is referenced by:  occon2t 9156  ococint 9292  chsscon3 9379  shjshs 9410

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-hilex 8864

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-oc 9119

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