Theorem shslub 9353

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 9342	. . . 4 |- (A (_ C -> (A +H B) (_ (C +H B))
5	2, 3	shscom 9327	. . . 4 |- (C +H B) = (B +H C)
6	4, 5	syl6ss 2110	. . 3 |- (A (_ C -> (A +H B) (_ (B +H C))
7	3, 2, 2	shless 9342	. . . 4 |- (B (_ C -> (B +H C) (_ (C +H C))
8	2	shsidm 9352	. . . 4 |- (C +H C) = C
9	7, 8	syl6ss 2110	. . 3 |- (B (_ C -> (B +H C) (_ C)
10	6, 9	sylan9ss 2078	. 2 |- ((A (_ C /\ B (_ C) -> (A +H B) (_ C)
11	1, 3	shsub1 9336	. . . 4 |- A (_ (A +H B)
12		sstr 2075	. . . 4 |- ((A (_ (A +H B) /\ (A +H B) (_ C) -> A (_ C)
13	11, 12	mpan 697	. . 3 |- ((A +H B) (_ C -> A (_ C)
14	3, 1	shsub2 9337	. . . 4 |- B (_ (A +H B)
15		sstr 2075	. . . 4 |- ((B (_ (A +H B) /\ (A +H B) (_ C) -> B (_ C)
16	14, 15	mpan 697	. . 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 960   (_ wss 2050  (class class class)co 3969  SHcsh 8792   +H cph 8795

This theorem is referenced by:  shlesb1 9354  shsumval2 9355

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-hilex 8864  ax-hfvadd 8865  ax-hvcom 8866  ax-hvaddid 8869

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-oprab 3972  df-sh 9071  df-shsum 9268

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