Theorem spanss 9306

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 2069	. . . . . . 7 |- (A (_ B -> (B (_ x -> A (_ x))
2	1	a1d 12	. . . . . 6 |- (A (_ B -> (x e. SH -> (B (_ x -> A (_ x)))
3	2	r19.21aiv 1712	. . . . 5 |- (A (_ B -> A.x e. SH (B (_ x -> A (_ x))
4		ss2rab 2121	. . . . 5 |- ({x e. SH | B (_ x} (_ {x e. SH | A (_ x} <-> A.x e. SH (B (_ x -> A (_ x))
5	3, 4	sylibr 200	. . . 4 |- (A (_ B -> {x e. SH | B (_ x} (_ {x e. SH | A (_ x})
6		intss 2551	. . . 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 10	. . 3 |- (A (_ B -> |^|{x e. SH | A (_ x} (_ |^|{x e. SH | B (_ x})
8	7	adantl 388	. 2 |- ((B (_ H~ /\ A (_ B) -> |^|{x e. SH | A (_ x} (_ |^|{x e. SH | B (_ x})
9		sstr 2070	. . . 4 |- ((A (_ B /\ B (_ H~) -> A (_ H~)
10	9	ancoms 436	. . 3 |- ((B (_ H~ /\ A (_ B) -> A (_ H~)
11		spanvalt 9287	. . 3 |- (A (_ H~ -> (span` A) = |^|{x e. SH | A (_ x})
12	10, 11	syl 10	. 2 |- ((B (_ H~ /\ A (_ B) -> (span` A) = |^|{x e. SH | A (_ x})
13		spanvalt 9287	. . 3 |- (B (_ H~ -> (span` B) = |^|{x e. SH | B (_ x})
14	13	adantr 389	. 2 |- ((B (_ H~ /\ A (_ B) -> (span` B) = |^|{x e. SH | B (_ x})
15	8, 12, 14	3sstr4d 2102	1 |- ((B (_ H~ /\ A (_ B) -> (span` A) (_ (span` B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1644  {crab 1647   (_ wss 2045  |^|cint 2530  ` cfv 3179  H~chil 8772  SHcsh 8781  spancspn 8785

This theorem is referenced by:  spanssoc 9307  span0 9453  spanun 9455  spansnpj 9491  shatomistic 10279

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-hilex 8853  ax-hfvadd 8854  ax-hv0cl 8857  ax-hfvmul 8859

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-rex 1649  df-rab 1651  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-int 2531  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-fv 3195  df-opr 3962  df-hlim 8825  df-sh 9064  df-ch 9080  df-span 9262

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