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 29059

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 28775	. . . . . . . . 9 ⊢ (⊥‘𝐴) ∈ C_ℋ
4	1, 3	chub2i 28938	. . . . . . . 8 ⊢ 𝐵 ⊆ ((⊥‘𝐴) ∨_ℋ 𝐵)
5		sseqin2 4112	. . . . . . . 8 ⊢ (𝐵 ⊆ ((⊥‘𝐴) ∨_ℋ 𝐵) ↔ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵)
6	4, 5	mpbi 231	. . . . . . 7 ⊢ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵
7	6	ineq2i 4106	. . . . . 6 ⊢ (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵)) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ 𝐵)
8		inass 4116	. . . . . 6 ⊢ ((((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) ∩ 𝐵) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵))
9	2, 1	chdmm1i 28945	. . . . . . 7 ⊢ (⊥‘(𝐴 ∩ 𝐵)) = ((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))
10	9	ineq1i 4105	. . . . . 6 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ 𝐵)
11	7, 8, 10	3eqtr4ri 2830	. . . . 5 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = ((((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) ∩ 𝐵)
12	2, 1	chdmj4i 28952	. . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) = (𝐴 ∩ 𝐵)
13	2, 1	chdmj2i 28950	. . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵)) = (𝐴 ∩ (⊥‘𝐵))
14	12, 13	oveq12i 7028	. . . . . . . . . 10 ⊢ ((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵))) = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵)))
15	14	eqeq2i 2807	. . . . . . . . 9 ⊢ (𝐴 = ((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵))) ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))))
16	15	biimpri 229	. . . . . . . 8 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → 𝐴 = ((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵))))
17	16	fveq2d 6542	. . . . . . 7 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → (⊥‘𝐴) = (⊥‘((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵)))))
18	1	choccli 28775	. . . . . . . . 9 ⊢ (⊥‘𝐵) ∈ C_ℋ
19	3, 18	chjcli 28925	. . . . . . . 8 ⊢ ((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∈ C_ℋ
20	3, 1	chjcli 28925	. . . . . . . 8 ⊢ ((⊥‘𝐴) ∨_ℋ 𝐵) ∈ C_ℋ
21	19, 20	chdmj4i 28952	. . . . . . 7 ⊢ (⊥‘((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵)))) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵))
22	17, 21	syl6req 2848	. . . . . 6 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) = (⊥‘𝐴))
23	22	ineq1d 4108	. . . . 5 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → ((((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) ∩ 𝐵) = ((⊥‘𝐴) ∩ 𝐵))
24	11, 23	syl5eq 2843	. . . 4 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = ((⊥‘𝐴) ∩ 𝐵))
25	24	oveq2d 7032	. . 3 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵)) = ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘𝐴) ∩ 𝐵)))
26		inss2 4126	. . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
27	2, 1	chincli 28928	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
28	27, 1	pjoml2i 29053	. . . 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 4099	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
31		incom 4099	. . . 4 ⊢ ((⊥‘𝐴) ∩ 𝐵) = (𝐵 ∩ (⊥‘𝐴))
32	30, 31	oveq12i 7028	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘𝐴) ∩ 𝐵)) = ((𝐵 ∩ 𝐴) ∨_ℋ (𝐵 ∩ (⊥‘𝐴)))
33	25, 29, 32	3eqtr3g 2854	. 2 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → 𝐵 = ((𝐵 ∩ 𝐴) ∨_ℋ (𝐵 ∩ (⊥‘𝐴))))
34	2, 1	cmbri 29058	. 2 ⊢ (𝐴 𝐶_ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))))
35	1, 2	cmbri 29058	. 2 ⊢ (𝐵 𝐶_ℋ 𝐴 ↔ 𝐵 = ((𝐵 ∩ 𝐴) ∨_ℋ (𝐵 ∩ (⊥‘𝐴))))
36	33, 34, 35	3imtr4i 293	1 ⊢ (𝐴 𝐶_ℋ 𝐵 → 𝐵 𝐶_ℋ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1522 ∈ wcel 2081 ∩ cin 3858 ⊆ wss 3859 class class class wbr 4962 ‘cfv 6225 (class class class)co 7016 C_ℋ cch 28397 ⊥cort 28398 ∨_ℋ chj 28401 𝐶_ℋ ccm 28404

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 ax-cc 9703 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 ax-hilex 28467 ax-hfvadd 28468 ax-hvcom 28469 ax-hvass 28470 ax-hv0cl 28471 ax-hvaddid 28472 ax-hfvmul 28473 ax-hvmulid 28474 ax-hvmulass 28475 ax-hvdistr1 28476 ax-hvdistr2 28477 ax-hvmul0 28478 ax-hfi 28547 ax-his1 28550 ax-his2 28551 ax-his3 28552 ax-his4 28553 ax-hcompl 28670

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-isom 6234 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-supp 7682 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-2o 7954 df-oadd 7957 df-omul 7958 df-er 8139 df-map 8258 df-pm 8259 df-ixp 8311 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-fsupp 8680 df-fi 8721 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 df-acn 9217 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-q 12198 df-rp 12240 df-xneg 12357 df-xadd 12358 df-xmul 12359 df-ioo 12592 df-ico 12594 df-icc 12595 df-fz 12743 df-fzo 12884 df-fl 13012 df-seq 13220 df-exp 13280 df-hash 13541 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-clim 14679 df-rlim 14680 df-sum 14877 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-starv 16409 df-sca 16410 df-vsca 16411 df-ip 16412 df-tset 16413 df-ple 16414 df-ds 16416 df-unif 16417 df-hom 16418 df-cco 16419 df-rest 16525 df-topn 16526 df-0g 16544 df-gsum 16545 df-topgen 16546 df-pt 16547 df-prds 16550 df-xrs 16604 df-qtop 16609 df-imas 16610 df-xps 16612 df-mre 16686 df-mrc 16687 df-acs 16689 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-mulg 17982 df-cntz 18188 df-cmn 18635 df-psmet 20219 df-xmet 20220 df-met 20221 df-bl 20222 df-mopn 20223 df-fbas 20224 df-fg 20225 df-cnfld 20228 df-top 21186 df-topon 21203 df-topsp 21225 df-bases 21238 df-cld 21311 df-ntr 21312 df-cls 21313 df-nei 21390 df-cn 21519 df-cnp 21520 df-lm 21521 df-haus 21607 df-tx 21854 df-hmeo 22047 df-fil 22138 df-fm 22230 df-flim 22231 df-flf 22232 df-xms 22613 df-ms 22614 df-tms 22615 df-cfil 23541 df-cau 23542 df-cmet 23543 df-grpo 27961 df-gid 27962 df-ginv 27963 df-gdiv 27964 df-ablo 28013 df-vc 28027 df-nv 28060 df-va 28063 df-ba 28064 df-sm 28065 df-0v 28066 df-vs 28067 df-nmcv 28068 df-ims 28069 df-dip 28169 df-ssp 28190 df-ph 28281 df-cbn 28331 df-hnorm 28436 df-hba 28437 df-hvsub 28439 df-hlim 28440 df-hcau 28441 df-sh 28675 df-ch 28689 df-oc 28720 df-ch0 28721 df-shs 28776 df-chj 28778 df-cm 29051

This theorem is referenced by: cmcmi 29060

W3C validator