Theorem occont 9160

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 2063	. . . . . . . . . 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 1289	. . . . . . . 8 |- (A (_ B -> (A.y(y e. B -> (x .ih y) = 0) -> A.y(y e. A -> (x .ih y) = 0)))
4		df-ral 1649	. . . . . . . 8 |- (A.y e. B (x .ih y) = 0 <-> A.y(y e. B -> (x .ih y) = 0))
5		df-ral 1649	. . . . . . . 8 |- (A.y e. A (x .ih y) = 0 <-> A.y(y e. A -> (x .ih y) = 0))
6	3, 4, 5	3imtr4g 553	. . . . . . 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 1713	. . . . 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 2123	. . . . 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 388	. . 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 9153	. . . 4 |- (B (_ H~ -> (_|_` B) = {x e. H~ | A.y e. B (x .ih y) = 0})
13	12	ad2antlr 405	. . 3 |- (((A (_ H~ /\ B (_ H~) /\ A (_ B) -> (_|_` B) = {x e. H~ | A.y e. B (x .ih y) = 0})
14		ocvalt 9153	. . . 4 |- (A (_ H~ -> (_|_` A) = {x e. H~ | A.y e. A (x .ih y) = 0})
15	14	ad2antrr 404	. . 3 |- (((A (_ H~ /\ B (_ H~) /\ A (_ B) -> (_|_` A) = {x e. H~ | A.y e. A (x .ih y) = 0})
16	11, 13, 15	3sstr4d 2104	. 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 954   = wceq 956   e. wcel 958  A.wral 1645  {crab 1648   (_ wss 2047  ` cfv 3182  (class class class)co 3963  0cc0 5234  H~chil 8788   .ih csp 8793  _|_cort 8799

This theorem is referenced by:  occon2t 9161  ococint 9297  chsscon3 9384  shjshs 9415

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-hilex 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-oc 9124

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