cmcmlem - Hilbert Space Explorer

	Hilbert Space Explorer		< Previous Next > Nearby theorems
Mirrors > Home > HSE Home > Th. List > cmcmlem		Structured version Visualization version GIF version

Theorem cmcmlem 29478

Description: Commutation is symmetric. Theorem 3.4 of [Beran] p. 45. (Contributed by NM, 3-Nov-2000.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
pjoml2.1	⊢ 𝐴 ∈ C_ℋ
pjoml2.2	⊢ 𝐵 ∈ C_ℋ

**Assertion**
Ref	Expression
cmcmlem	⊢ (𝐴 𝐶_ℋ 𝐵 → 𝐵 𝐶_ℋ 𝐴)

**Proof of Theorem cmcmlem**
Step	Hyp	Ref	Expression
1		pjoml2.2	. . . . . . . . 9 ⊢ 𝐵 ∈ C_ℋ
2		pjoml2.1	. . . . . . . . . 10 ⊢ 𝐴 ∈ C_ℋ
3	2	choccli 29194	. . . . . . . . 9 ⊢ (⊥‘𝐴) ∈ C_ℋ
4	1, 3	chub2i 29357	. . . . . . . 8 ⊢ 𝐵 ⊆ ((⊥‘𝐴) ∨_ℋ 𝐵)
5		sseqin2 4122	. . . . . . . 8 ⊢ (𝐵 ⊆ ((⊥‘𝐴) ∨_ℋ 𝐵) ↔ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵)
6	4, 5	mpbi 233	. . . . . . 7 ⊢ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵
7	6	ineq2i 4116	. . . . . 6 ⊢ (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵)) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ 𝐵)
8		inass 4126	. . . . . 6 ⊢ ((((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) ∩ 𝐵) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵))
9	2, 1	chdmm1i 29364	. . . . . . 7 ⊢ (⊥‘(𝐴 ∩ 𝐵)) = ((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))
10	9	ineq1i 4115	. . . . . 6 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ 𝐵)
11	7, 8, 10	3eqtr4ri 2792	. . . . 5 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = ((((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) ∩ 𝐵)
12	2, 1	chdmj4i 29371	. . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) = (𝐴 ∩ 𝐵)
13	2, 1	chdmj2i 29369	. . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵)) = (𝐴 ∩ (⊥‘𝐵))
14	12, 13	oveq12i 7167	. . . . . . . . . 10 ⊢ ((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵))) = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵)))
15	14	eqeq2i 2771	. . . . . . . . 9 ⊢ (𝐴 = ((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵))) ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))))
16	15	biimpri 231	. . . . . . . 8 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → 𝐴 = ((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵))))
17	16	fveq2d 6666	. . . . . . 7 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → (⊥‘𝐴) = (⊥‘((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵)))))
18	1	choccli 29194	. . . . . . . . 9 ⊢ (⊥‘𝐵) ∈ C_ℋ
19	3, 18	chjcli 29344	. . . . . . . 8 ⊢ ((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∈ C_ℋ
20	3, 1	chjcli 29344	. . . . . . . 8 ⊢ ((⊥‘𝐴) ∨_ℋ 𝐵) ∈ C_ℋ
21	19, 20	chdmj4i 29371	. . . . . . 7 ⊢ (⊥‘((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵)))) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵))
22	17, 21	eqtr2di 2810	. . . . . 6 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) = (⊥‘𝐴))
23	22	ineq1d 4118	. . . . 5 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → ((((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) ∩ 𝐵) = ((⊥‘𝐴) ∩ 𝐵))
24	11, 23	syl5eq 2805	. . . 4 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = ((⊥‘𝐴) ∩ 𝐵))
25	24	oveq2d 7171	. . 3 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵)) = ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘𝐴) ∩ 𝐵)))
26		inss2 4136	. . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
27	2, 1	chincli 29347	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
28	27, 1	pjoml2i 29472	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ⊆ 𝐵 → ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵)) = 𝐵)
29	26, 28	ax-mp 5	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵)) = 𝐵
30		incom 4108	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
31		incom 4108	. . . 4 ⊢ ((⊥‘𝐴) ∩ 𝐵) = (𝐵 ∩ (⊥‘𝐴))
32	30, 31	oveq12i 7167	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘𝐴) ∩ 𝐵)) = ((𝐵 ∩ 𝐴) ∨_ℋ (𝐵 ∩ (⊥‘𝐴)))
33	25, 29, 32	3eqtr3g 2816	. 2 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → 𝐵 = ((𝐵 ∩ 𝐴) ∨_ℋ (𝐵 ∩ (⊥‘𝐴))))
34	2, 1	cmbri 29477	. 2 ⊢ (𝐴 𝐶_ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))))
35	1, 2	cmbri 29477	. 2 ⊢ (𝐵 𝐶_ℋ 𝐴 ↔ 𝐵 = ((𝐵 ∩ 𝐴) ∨_ℋ (𝐵 ∩ (⊥‘𝐴))))
36	33, 34, 35	3imtr4i 295	1 ⊢ (𝐴 𝐶_ℋ 𝐵 → 𝐵 𝐶_ℋ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∩ cin 3859 ⊆ wss 3860 class class class wbr 5035 ‘cfv 6339 (class class class)co 7155 C_ℋ cch 28816 ⊥cort 28817 ∨_ℋ chj 28820 𝐶_ℋ ccm 28823

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5159 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-inf2 9142 ax-cc 9900 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656 ax-pre-mulgt0 10657 ax-pre-sup 10658 ax-addf 10659 ax-mulf 10660 ax-hilex 28886 ax-hfvadd 28887 ax-hvcom 28888 ax-hvass 28889 ax-hv0cl 28890 ax-hvaddid 28891 ax-hfvmul 28892 ax-hvmulid 28893 ax-hvmulass 28894 ax-hvdistr1 28895 ax-hvdistr2 28896 ax-hvmul0 28897 ax-hfi 28966 ax-his1 28969 ax-his2 28970 ax-his3 28971 ax-his4 28972 ax-hcompl 29089

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-int 4842 df-iun 4888 df-iin 4889 df-br 5036 df-opab 5098 df-mpt 5116 df-tr 5142 df-id 5433 df-eprel 5438 df-po 5446 df-so 5447 df-fr 5486 df-se 5487 df-we 5488 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7410 df-om 7585 df-1st 7698 df-2nd 7699 df-supp 7841 df-wrecs 7962 df-recs 8023 df-rdg 8061 df-1o 8117 df-2o 8118 df-oadd 8121 df-omul 8122 df-er 8304 df-map 8423 df-pm 8424 df-ixp 8485 df-en 8533 df-dom 8534 df-sdom 8535 df-fin 8536 df-fsupp 8872 df-fi 8913 df-sup 8944 df-inf 8945 df-oi 9012 df-card 9406 df-acn 9409 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-div 11341 df-nn 11680 df-2 11742 df-3 11743 df-4 11744 df-5 11745 df-6 11746 df-7 11747 df-8 11748 df-9 11749 df-n0 11940 df-z 12026 df-dec 12143 df-uz 12288 df-q 12394 df-rp 12436 df-xneg 12553 df-xadd 12554 df-xmul 12555 df-ioo 12788 df-ico 12790 df-icc 12791 df-fz 12945 df-fzo 13088 df-fl 13216 df-seq 13424 df-exp 13485 df-hash 13746 df-cj 14511 df-re 14512 df-im 14513 df-sqrt 14647 df-abs 14648 df-clim 14898 df-rlim 14899 df-sum 15096 df-struct 16548 df-ndx 16549 df-slot 16550 df-base 16552 df-sets 16553 df-ress 16554 df-plusg 16641 df-mulr 16642 df-starv 16643 df-sca 16644 df-vsca 16645 df-ip 16646 df-tset 16647 df-ple 16648 df-ds 16650 df-unif 16651 df-hom 16652 df-cco 16653 df-rest 16759 df-topn 16760 df-0g 16778 df-gsum 16779 df-topgen 16780 df-pt 16781 df-prds 16784 df-xrs 16838 df-qtop 16843 df-imas 16844 df-xps 16846 df-mre 16920 df-mrc 16921 df-acs 16923 df-mgm 17923 df-sgrp 17972 df-mnd 17983 df-submnd 18028 df-mulg 18297 df-cntz 18519 df-cmn 18980 df-psmet 20163 df-xmet 20164 df-met 20165 df-bl 20166 df-mopn 20167 df-fbas 20168 df-fg 20169 df-cnfld 20172 df-top 21599 df-topon 21616 df-topsp 21638 df-bases 21651 df-cld 21724 df-ntr 21725 df-cls 21726 df-nei 21803 df-cn 21932 df-cnp 21933 df-lm 21934 df-haus 22020 df-tx 22267 df-hmeo 22460 df-fil 22551 df-fm 22643 df-flim 22644 df-flf 22645 df-xms 23027 df-ms 23028 df-tms 23029 df-cfil 23960 df-cau 23961 df-cmet 23962 df-grpo 28380 df-gid 28381 df-ginv 28382 df-gdiv 28383 df-ablo 28432 df-vc 28446 df-nv 28479 df-va 28482 df-ba 28483 df-sm 28484 df-0v 28485 df-vs 28486 df-nmcv 28487 df-ims 28488 df-dip 28588 df-ssp 28609 df-ph 28700 df-cbn 28750 df-hnorm 28855 df-hba 28856 df-hvsub 28858 df-hlim 28859 df-hcau 28860 df-sh 29094 df-ch 29108 df-oc 29139 df-ch0 29140 df-shs 29195 df-chj 29197 df-cm 29470

This theorem is referenced by: cmcmi 29479

W3C validator