Theorem spanss 5319

Description: Ordering relationship for the spans of subsets of Hilbert space.

Assertion

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

Proof of Theorem spanss

Step	Hyp	Ref	Expression
1		sstr2 1510	. . . . . . 7 |- (A (_ B -> (B (_ x -> A (_ x))
2	1	a1d 14	. . . . . 6 |- (A (_ B -> (x e. SH -> (B (_ x -> A (_ x)))
3	2	r19.21aiv 1259	. . . . 5 |- (A (_ B -> A.x e. SH (B (_ x -> A (_ x))
4		ss2rab 1553	. . . . 5 |- ({x e. SH | B (_ x} (_ {x e. SH | A (_ x} <-> A.x e. SH (B (_ x -> A (_ x))
5	3, 4	sylibr 175	. . . 4 |- (A (_ B -> {x e. SH | B (_ x} (_ {x e. SH | A (_ x})
6		intss 1983	. . . 4 |- ({x e. SH | B (_ x} (_ {x e. SH | A (_ x} -> |^|{x e. SH | A (_ x} (_ |^|{x e. SH | B (_ x})
7	5, 6	syl 12	. . 3 |- (A (_ B -> |^|{x e. SH | A (_ x} (_ |^|{x e. SH | B (_ x})
8	7	adantl 305	. 2 |- ((B (_ H~ /\ A (_ B) -> |^|{x e. SH | A (_ x} (_ |^|{x e. SH | B (_ x})
9		sstr 1511	. . . 4 |- ((A (_ B /\ B (_ H~) -> A (_ H~)
10	9	ancoms 334	. . 3 |- ((B (_ H~ /\ A (_ B) -> A (_ H~)
11		spanvalt 5300	. . 3 |- (A (_ H~ -> (span` A) = |^|{x e. SH | A (_ x})
12	10, 11	syl 12	. 2 |- ((B (_ H~ /\ A (_ B) -> (span` A) = |^|{x e. SH | A (_ x})
13		spanvalt 5300	. . 3 |- (B (_ H~ -> (span` B) = |^|{x e. SH | B (_ x})
14	13	adantr 306	. 2 |- ((B (_ H~ /\ A (_ B) -> (span` B) = |^|{x e. SH | B (_ x})
15	8, 12, 14	3sstr4d 1543	1 |- ((B (_ H~ /\ A (_ B) -> (span` A) (_ (span` B))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092  A.wral 1201  {crab 1204   (_ wss 1487  |^|cint 1965  ` cfv 2422  H~chil 4958  SHcsh 4967  spancspn 4971

This theorem is referenced by:  spanssoc 5320  span0 5448  spanun 5450  spansnpj 5481

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  ax-hvaddcl 4984  ax-hvzercl 4987  ax-hvmulcl 4989

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-int 1966  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-fn 2433  df-f 2434  df-fv 2438  df-opr 3003  df-hlim 5107  df-sh 5114  df-ch 5127  df-span 5276

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