Theorem shslub 9358

Description: Least upper bound law for Hilbert subspace sum.

Hypotheses

Ref	Expression
shslub.1	|- A e. SH
shslub.2	|- B e. SH
shslub.3	|- C e. SH

Assertion

Ref	Expression
shslub	|- ((A (_ C /\ B (_ C) <-> (A +H B) (_ C)

Proof of Theorem shslub

Step	Hyp	Ref	Expression
1		shslub.1	. . . . 5 |- A e. SH
2		shslub.3	. . . . 5 |- C e. SH
3		shslub.2	. . . . 5 |- B e. SH
4	1, 2, 3	shless 9347	. . . 4 |- (A (_ C -> (A +H B) (_ (C +H B))
5	2, 3	shscom 9332	. . . 4 |- (C +H B) = (B +H C)
6	4, 5	syl6ss 2107	. . 3 |- (A (_ C -> (A +H B) (_ (B +H C))
7	3, 2, 2	shless 9347	. . . 4 |- (B (_ C -> (B +H C) (_ (C +H C))
8	2	shsidm 9357	. . . 4 |- (C +H C) = C
9	7, 8	syl6ss 2107	. . 3 |- (B (_ C -> (B +H C) (_ C)
10	6, 9	sylan9ss 2075	. 2 |- ((A (_ C /\ B (_ C) -> (A +H B) (_ C)
11	1, 3	shsub1 9341	. . . 4 |- A (_ (A +H B)
12		sstr 2072	. . . 4 |- ((A (_ (A +H B) /\ (A +H B) (_ C) -> A (_ C)
13	11, 12	mpan 695	. . 3 |- ((A +H B) (_ C -> A (_ C)
14	3, 1	shsub2 9342	. . . 4 |- B (_ (A +H B)
15		sstr 2072	. . . 4 |- ((B (_ (A +H B) /\ (A +H B) (_ C) -> B (_ C)
16	14, 15	mpan 695	. . 3 |- ((A +H B) (_ C -> B (_ C)
17	13, 16	jca 288	. 2 |- ((A +H B) (_ C -> (A (_ C /\ B (_ C))
18	10, 17	impbi 157	1 |- ((A (_ C /\ B (_ C) <-> (A +H B) (_ C)

Syntax hints:   <-> wb 146   /\ wa 223   e. wcel 958   (_ wss 2047  (class class class)co 3963  SHcsh 8797   +H cph 8800

This theorem is referenced by:  shlesb1 9359  shsumval2 9360

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-un 2866  ax-hilex 8869  ax-hfvadd 8870  ax-hvcom 8871  ax-hvaddid 8874

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-sbc 1942  df-csb 2002  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-fn 3193  df-f 3194  df-fv 3198  df-opr 3965  df-oprab 3966  df-sh 9076  df-shsum 9273

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