Theorem shslub 5359

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 5348	. . . 4 |- (A (_ C -> (A +H B) (_ (C +H B))
5	2, 3	shscom 5333	. . . 4 |- (C +H B) = (B +H C)
6	4, 5	syl6ss 1546	. . 3 |- (A (_ C -> (A +H B) (_ (B +H C))
7	3, 2, 2	shless 5348	. . . 4 |- (B (_ C -> (B +H C) (_ (C +H C))
8	2	shsidm 5358	. . . 4 |- (C +H C) = C
9	7, 8	syl6ss 1546	. . 3 |- (B (_ C -> (B +H C) (_ C)
10	6, 9	sylan9ss 1514	. 2 |- ((A (_ C /\ B (_ C) -> (A +H B) (_ C)
11	1, 3	shsub1 5342	. . . 4 |- A (_ (A +H B)
12		sstr 1511	. . . 4 |- ((A (_ (A +H B) /\ (A +H B) (_ C) -> A (_ C)
13	11, 12	mpan 518	. . 3 |- ((A +H B) (_ C -> A (_ C)
14	3, 1	shsub1 5342	. . . . 5 |- B (_ (B +H A)
15	3, 1	shscom 5333	. . . . 5 |- (B +H A) = (A +H B)
16	14, 15	sseqtr 1532	. . . 4 |- B (_ (A +H B)
17		sstr 1511	. . . 4 |- ((B (_ (A +H B) /\ (A +H B) (_ C) -> B (_ C)
18	16, 17	mpan 518	. . 3 |- ((A +H B) (_ C -> B (_ C)
19	13, 18	jca 236	. 2 |- ((A +H B) (_ C -> (A (_ C /\ B (_ C))
20	10, 19	impbi 139	1 |- ((A (_ C /\ B (_ C) <-> (A +H B) (_ C)

Syntax hints:   <-> wb 127   /\ wa 196   e. wcel 1092   (_ wss 1487  (class class class)co 3001  SHcsh 4967   +H cph 4970

This theorem is referenced by:  shlesb1 5360  shsumval2 5361

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-hvcom 4985  ax-hvaddid 4988

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  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-opr 3003  df-oprab 3004  df-sh 5114  df-shsum 5275

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