Theorem occont 5168

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 1502	. . . . . . . . . 10 ⊢ (A ⊆ B → (y ∈ A → y ∈ B))
2	1	syl4d 28	. . . . . . . . 9 ⊢ (A ⊆ B → ((y ∈ B → (x ·i y) = 0) → (y ∈ A → (x ·i y) = 0)))
3	2	19.20dv 946	. . . . . . . 8 ⊢ (A ⊆ B → (∀y(y ∈ B → (x ·i y) = 0) → ∀y(y ∈ A → (x ·i y) = 0)))
4		df-ral 1205	. . . . . . . 8 ⊢ (∀y ∈ B (x ·i y) = 0 ↔ ∀y(y ∈ B → (x ·i y) = 0))
5		df-ral 1205	. . . . . . . 8 ⊢ (∀y ∈ A (x ·i y) = 0 ↔ ∀y(y ∈ A → (x ·i y) = 0))
6	3, 4, 5	3imtr4g 426	. . . . . . 7 ⊢ (A ⊆ B → (∀y ∈ B (x ·i y) = 0 → ∀y ∈ A (x ·i y) = 0))
7	6	a1d 14	. . . . . 6 ⊢ (A ⊆ B → (x ∈ ℋ → (∀y ∈ B (x ·i y) = 0 → ∀y ∈ A (x ·i y) = 0)))
8	7	r19.21aiv 1259	. . . . 5 ⊢ (A ⊆ B → ∀x ∈ ℋ (∀y ∈ B (x ·i y) = 0 → ∀y ∈ A (x ·i y) = 0))
9		ss2rab 1553	. . . . 5 ⊢ ({x ∈ ℋ ∣∀y ∈ B (x ·i y) = 0} ⊆ {x ∈ ℋ ∣∀y ∈ A (x ·i y) = 0} ↔ ∀x ∈ ℋ (∀y ∈ B (x ·i y) = 0 → ∀y ∈ A (x ·i y) = 0))
10	8, 9	sylibr 175	. . . 4 ⊢ (A ⊆ B → {x ∈ ℋ ∣∀y ∈ B (x ·i y) = 0} ⊆ {x ∈ ℋ ∣∀y ∈ A (x ·i y) = 0})
11	10	adantl 305	. . 3 ⊢ (((A ⊆ ℋ ∧ B ⊆ ℋ ) ∧ A ⊆ B) → {x ∈ ℋ ∣∀y ∈ B (x ·i y) = 0} ⊆ {x ∈ ℋ ∣∀y ∈ A (x ·i y) = 0})
12		ocvalt 5161	. . . 4 ⊢ (B ⊆ ℋ → (⊥ ‘B) = {x ∈ ℋ ∣∀y ∈ B (x ·i y) = 0})
13	12	ad2antlr 321	. . 3 ⊢ (((A ⊆ ℋ ∧ B ⊆ ℋ ) ∧ A ⊆ B) → (⊥ ‘B) = {x ∈ ℋ ∣∀y ∈ B (x ·i y) = 0})
14		ocvalt 5161	. . . 4 ⊢ (A ⊆ ℋ → (⊥ ‘A) = {x ∈ ℋ ∣∀y ∈ A (x ·i y) = 0})
15	14	ad2antll 320	. . 3 ⊢ (((A ⊆ ℋ ∧ B ⊆ ℋ ) ∧ A ⊆ B) → (⊥ ‘A) = {x ∈ ℋ ∣∀y ∈ A (x ·i y) = 0})
16	11, 13, 15	3sstr4d 1543	. 2 ⊢ (((A ⊆ ℋ ∧ B ⊆ ℋ ) ∧ A ⊆ B) → (⊥ ‘B) ⊆ (⊥ ‘A))
17	16	exp 291	1 ⊢ ((A ⊆ ℋ ∧ B ⊆ ℋ ) → (A ⊆ B → (⊥ ‘B) ⊆ (⊥ ‘A)))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204   ⊆ wss 1487   ‘cfv 2422  (class class class)co 3001  0cc0 4028   ℋ chil 4958   ·_i csp 4963  ⊥cort 4969

This theorem is referenced by:  occon2t 5169  ococint 5298  chsscon3 5383  shjshs 5412

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077  ax-hilex 4983

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-oc 5156

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