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Theorem cmcmlem 31678

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 31394	. . . . . . . . 9 ⊢ (⊥‘𝐴) ∈ C_ℋ
4	1, 3	chub2i 31557	. . . . . . . 8 ⊢ 𝐵 ⊆ ((⊥‘𝐴) ∨_ℋ 𝐵)
5		sseqin2 4177	. . . . . . . 8 ⊢ (𝐵 ⊆ ((⊥‘𝐴) ∨_ℋ 𝐵) ↔ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵)
6	4, 5	mpbi 230	. . . . . . 7 ⊢ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵) = 𝐵
7	6	ineq2i 4171	. . . . . 6 ⊢ (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵)) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ 𝐵)
8		inass 4182	. . . . . 6 ⊢ ((((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) ∩ 𝐵) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ (((⊥‘𝐴) ∨_ℋ 𝐵) ∩ 𝐵))
9	2, 1	chdmm1i 31564	. . . . . . 7 ⊢ (⊥‘(𝐴 ∩ 𝐵)) = ((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))
10	9	ineq1i 4170	. . . . . 6 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ 𝐵)
11	7, 8, 10	3eqtr4ri 2771	. . . . 5 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = ((((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) ∩ 𝐵)
12	2, 1	chdmj4i 31571	. . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) = (𝐴 ∩ 𝐵)
13	2, 1	chdmj2i 31569	. . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵)) = (𝐴 ∩ (⊥‘𝐵))
14	12, 13	oveq12i 7380	. . . . . . . . . 10 ⊢ ((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵))) = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵)))
15	14	eqeq2i 2750	. . . . . . . . 9 ⊢ (𝐴 = ((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵))) ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))))
16	15	biimpri 228	. . . . . . . 8 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → 𝐴 = ((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵))))
17	16	fveq2d 6846	. . . . . . 7 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → (⊥‘𝐴) = (⊥‘((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵)))))
18	1	choccli 31394	. . . . . . . . 9 ⊢ (⊥‘𝐵) ∈ C_ℋ
19	3, 18	chjcli 31544	. . . . . . . 8 ⊢ ((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∈ C_ℋ
20	3, 1	chjcli 31544	. . . . . . . 8 ⊢ ((⊥‘𝐴) ∨_ℋ 𝐵) ∈ C_ℋ
21	19, 20	chdmj4i 31571	. . . . . . 7 ⊢ (⊥‘((⊥‘((⊥‘𝐴) ∨_ℋ (⊥‘𝐵))) ∨_ℋ (⊥‘((⊥‘𝐴) ∨_ℋ 𝐵)))) = (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵))
22	17, 21	eqtr2di 2789	. . . . . 6 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → (((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) = (⊥‘𝐴))
23	22	ineq1d 4173	. . . . 5 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → ((((⊥‘𝐴) ∨_ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨_ℋ 𝐵)) ∩ 𝐵) = ((⊥‘𝐴) ∩ 𝐵))
24	11, 23	eqtrid 2784	. . . 4 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = ((⊥‘𝐴) ∩ 𝐵))
25	24	oveq2d 7384	. . 3 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵)) = ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘𝐴) ∩ 𝐵)))
26		inss2 4192	. . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
27	2, 1	chincli 31547	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ∈ C_ℋ
28	27, 1	pjoml2i 31672	. . . 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 4163	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
31		incom 4163	. . . 4 ⊢ ((⊥‘𝐴) ∩ 𝐵) = (𝐵 ∩ (⊥‘𝐴))
32	30, 31	oveq12i 7380	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∨_ℋ ((⊥‘𝐴) ∩ 𝐵)) = ((𝐵 ∩ 𝐴) ∨_ℋ (𝐵 ∩ (⊥‘𝐴)))
33	25, 29, 32	3eqtr3g 2795	. 2 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))) → 𝐵 = ((𝐵 ∩ 𝐴) ∨_ℋ (𝐵 ∩ (⊥‘𝐴))))
34	2, 1	cmbri 31677	. 2 ⊢ (𝐴 𝐶_ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨_ℋ (𝐴 ∩ (⊥‘𝐵))))
35	1, 2	cmbri 31677	. 2 ⊢ (𝐵 𝐶_ℋ 𝐴 ↔ 𝐵 = ((𝐵 ∩ 𝐴) ∨_ℋ (𝐵 ∩ (⊥‘𝐴))))
36	33, 34, 35	3imtr4i 292	1 ⊢ (𝐴 𝐶_ℋ 𝐵 → 𝐵 𝐶_ℋ 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∩ cin 3902 ⊆ wss 3903 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 C_ℋ cch 31016 ⊥cort 31017 ∨_ℋ chj 31020 𝐶_ℋ ccm 31023

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cc 10357 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 ax-hilex 31086 ax-hfvadd 31087 ax-hvcom 31088 ax-hvass 31089 ax-hv0cl 31090 ax-hvaddid 31091 ax-hfvmul 31092 ax-hvmulid 31093 ax-hvmulass 31094 ax-hvdistr1 31095 ax-hvdistr2 31096 ax-hvmul0 31097 ax-hfi 31166 ax-his1 31169 ax-his2 31170 ax-his3 31171 ax-his4 31172 ax-hcompl 31289

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-omul 8412 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-acn 9866 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-rlim 15424 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-cn 23183 df-cnp 23184 df-lm 23185 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cfil 25223 df-cau 25224 df-cmet 25225 df-grpo 30580 df-gid 30581 df-ginv 30582 df-gdiv 30583 df-ablo 30632 df-vc 30646 df-nv 30679 df-va 30682 df-ba 30683 df-sm 30684 df-0v 30685 df-vs 30686 df-nmcv 30687 df-ims 30688 df-dip 30788 df-ssp 30809 df-ph 30900 df-cbn 30950 df-hnorm 31055 df-hba 31056 df-hvsub 31058 df-hlim 31059 df-hcau 31060 df-sh 31294 df-ch 31308 df-oc 31339 df-ch0 31340 df-shs 31395 df-chj 31397 df-cm 31670

This theorem is referenced by: cmcmi 31679

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