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Mirrors > Home > HSE Home > Th. List > cmcmlem | Structured version Visualization version GIF version |
Description: Commutation is symmetric. Theorem 3.4 of [Beran] p. 45. (Contributed by NM, 3-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjoml2.1 | ⊢ 𝐴 ∈ Cℋ |
pjoml2.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
cmcmlem | ⊢ (𝐴 𝐶ℋ 𝐵 → 𝐵 𝐶ℋ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjoml2.2 | . . . . . . . . 9 ⊢ 𝐵 ∈ Cℋ | |
2 | pjoml2.1 | . . . . . . . . . 10 ⊢ 𝐴 ∈ Cℋ | |
3 | 2 | choccli 29012 | . . . . . . . . 9 ⊢ (⊥‘𝐴) ∈ Cℋ |
4 | 1, 3 | chub2i 29175 | . . . . . . . 8 ⊢ 𝐵 ⊆ ((⊥‘𝐴) ∨ℋ 𝐵) |
5 | sseqin2 4191 | . . . . . . . 8 ⊢ (𝐵 ⊆ ((⊥‘𝐴) ∨ℋ 𝐵) ↔ (((⊥‘𝐴) ∨ℋ 𝐵) ∩ 𝐵) = 𝐵) | |
6 | 4, 5 | mpbi 231 | . . . . . . 7 ⊢ (((⊥‘𝐴) ∨ℋ 𝐵) ∩ 𝐵) = 𝐵 |
7 | 6 | ineq2i 4185 | . . . . . 6 ⊢ (((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ (((⊥‘𝐴) ∨ℋ 𝐵) ∩ 𝐵)) = (((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ 𝐵) |
8 | inass 4195 | . . . . . 6 ⊢ ((((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) ∩ 𝐵) = (((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ (((⊥‘𝐴) ∨ℋ 𝐵) ∩ 𝐵)) | |
9 | 2, 1 | chdmm1i 29182 | . . . . . . 7 ⊢ (⊥‘(𝐴 ∩ 𝐵)) = ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) |
10 | 9 | ineq1i 4184 | . . . . . 6 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = (((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ 𝐵) |
11 | 7, 8, 10 | 3eqtr4ri 2855 | . . . . 5 ⊢ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = ((((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) ∩ 𝐵) |
12 | 2, 1 | chdmj4i 29189 | . . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) = (𝐴 ∩ 𝐵) |
13 | 2, 1 | chdmj2i 29187 | . . . . . . . . . . 11 ⊢ (⊥‘((⊥‘𝐴) ∨ℋ 𝐵)) = (𝐴 ∩ (⊥‘𝐵)) |
14 | 12, 13 | oveq12i 7157 | . . . . . . . . . 10 ⊢ ((⊥‘((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) ∨ℋ (⊥‘((⊥‘𝐴) ∨ℋ 𝐵))) = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))) |
15 | 14 | eqeq2i 2834 | . . . . . . . . 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 6668 | . . . . . . 7 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))) → (⊥‘𝐴) = (⊥‘((⊥‘((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) ∨ℋ (⊥‘((⊥‘𝐴) ∨ℋ 𝐵))))) |
18 | 1 | choccli 29012 | . . . . . . . . 9 ⊢ (⊥‘𝐵) ∈ Cℋ |
19 | 3, 18 | chjcli 29162 | . . . . . . . 8 ⊢ ((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∈ Cℋ |
20 | 3, 1 | chjcli 29162 | . . . . . . . 8 ⊢ ((⊥‘𝐴) ∨ℋ 𝐵) ∈ Cℋ |
21 | 19, 20 | chdmj4i 29189 | . . . . . . 7 ⊢ (⊥‘((⊥‘((⊥‘𝐴) ∨ℋ (⊥‘𝐵))) ∨ℋ (⊥‘((⊥‘𝐴) ∨ℋ 𝐵)))) = (((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) |
22 | 17, 21 | syl6req 2873 | . . . . . 6 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))) → (((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) = (⊥‘𝐴)) |
23 | 22 | ineq1d 4187 | . . . . 5 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))) → ((((⊥‘𝐴) ∨ℋ (⊥‘𝐵)) ∩ ((⊥‘𝐴) ∨ℋ 𝐵)) ∩ 𝐵) = ((⊥‘𝐴) ∩ 𝐵)) |
24 | 11, 23 | syl5eq 2868 | . . . 4 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))) → ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵) = ((⊥‘𝐴) ∩ 𝐵)) |
25 | 24 | oveq2d 7161 | . . 3 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))) → ((𝐴 ∩ 𝐵) ∨ℋ ((⊥‘(𝐴 ∩ 𝐵)) ∩ 𝐵)) = ((𝐴 ∩ 𝐵) ∨ℋ ((⊥‘𝐴) ∩ 𝐵))) |
26 | inss2 4205 | . . . 4 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵 | |
27 | 2, 1 | chincli 29165 | . . . . 5 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
28 | 27, 1 | pjoml2i 29290 | . . . 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 4177 | . . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴) | |
31 | incom 4177 | . . . 4 ⊢ ((⊥‘𝐴) ∩ 𝐵) = (𝐵 ∩ (⊥‘𝐴)) | |
32 | 30, 31 | oveq12i 7157 | . . 3 ⊢ ((𝐴 ∩ 𝐵) ∨ℋ ((⊥‘𝐴) ∩ 𝐵)) = ((𝐵 ∩ 𝐴) ∨ℋ (𝐵 ∩ (⊥‘𝐴))) |
33 | 25, 29, 32 | 3eqtr3g 2879 | . 2 ⊢ (𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵))) → 𝐵 = ((𝐵 ∩ 𝐴) ∨ℋ (𝐵 ∩ (⊥‘𝐴)))) |
34 | 2, 1 | cmbri 29295 | . 2 ⊢ (𝐴 𝐶ℋ 𝐵 ↔ 𝐴 = ((𝐴 ∩ 𝐵) ∨ℋ (𝐴 ∩ (⊥‘𝐵)))) |
35 | 1, 2 | cmbri 29295 | . 2 ⊢ (𝐵 𝐶ℋ 𝐴 ↔ 𝐵 = ((𝐵 ∩ 𝐴) ∨ℋ (𝐵 ∩ (⊥‘𝐴)))) |
36 | 33, 34, 35 | 3imtr4i 293 | 1 ⊢ (𝐴 𝐶ℋ 𝐵 → 𝐵 𝐶ℋ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∩ cin 3934 ⊆ wss 3935 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 Cℋ cch 28634 ⊥cort 28635 ∨ℋ chj 28638 𝐶ℋ ccm 28641 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-cc 9846 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 ax-hilex 28704 ax-hfvadd 28705 ax-hvcom 28706 ax-hvass 28707 ax-hv0cl 28708 ax-hvaddid 28709 ax-hfvmul 28710 ax-hvmulid 28711 ax-hvmulass 28712 ax-hvdistr1 28713 ax-hvdistr2 28714 ax-hvmul0 28715 ax-hfi 28784 ax-his1 28787 ax-his2 28788 ax-his3 28789 ax-his4 28790 ax-hcompl 28907 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7569 df-1st 7680 df-2nd 7681 df-supp 7822 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-2o 8094 df-oadd 8097 df-omul 8098 df-er 8279 df-map 8398 df-pm 8399 df-ixp 8451 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-fsupp 8823 df-fi 8864 df-sup 8895 df-inf 8896 df-oi 8963 df-card 9357 df-acn 9360 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11628 df-2 11689 df-3 11690 df-4 11691 df-5 11692 df-6 11693 df-7 11694 df-8 11695 df-9 11696 df-n0 11887 df-z 11971 df-dec 12088 df-uz 12233 df-q 12338 df-rp 12380 df-xneg 12497 df-xadd 12498 df-xmul 12499 df-ioo 12732 df-ico 12734 df-icc 12735 df-fz 12883 df-fzo 13024 df-fl 13152 df-seq 13360 df-exp 13420 df-hash 13681 df-cj 14448 df-re 14449 df-im 14450 df-sqrt 14584 df-abs 14585 df-clim 14835 df-rlim 14836 df-sum 15033 df-struct 16475 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-ress 16481 df-plusg 16568 df-mulr 16569 df-starv 16570 df-sca 16571 df-vsca 16572 df-ip 16573 df-tset 16574 df-ple 16575 df-ds 16577 df-unif 16578 df-hom 16579 df-cco 16580 df-rest 16686 df-topn 16687 df-0g 16705 df-gsum 16706 df-topgen 16707 df-pt 16708 df-prds 16711 df-xrs 16765 df-qtop 16770 df-imas 16771 df-xps 16773 df-mre 16847 df-mrc 16848 df-acs 16850 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-submnd 17947 df-mulg 18165 df-cntz 18387 df-cmn 18839 df-psmet 20467 df-xmet 20468 df-met 20469 df-bl 20470 df-mopn 20471 df-fbas 20472 df-fg 20473 df-cnfld 20476 df-top 21432 df-topon 21449 df-topsp 21471 df-bases 21484 df-cld 21557 df-ntr 21558 df-cls 21559 df-nei 21636 df-cn 21765 df-cnp 21766 df-lm 21767 df-haus 21853 df-tx 22100 df-hmeo 22293 df-fil 22384 df-fm 22476 df-flim 22477 df-flf 22478 df-xms 22859 df-ms 22860 df-tms 22861 df-cfil 23787 df-cau 23788 df-cmet 23789 df-grpo 28198 df-gid 28199 df-ginv 28200 df-gdiv 28201 df-ablo 28250 df-vc 28264 df-nv 28297 df-va 28300 df-ba 28301 df-sm 28302 df-0v 28303 df-vs 28304 df-nmcv 28305 df-ims 28306 df-dip 28406 df-ssp 28427 df-ph 28518 df-cbn 28568 df-hnorm 28673 df-hba 28674 df-hvsub 28676 df-hlim 28677 df-hcau 28678 df-sh 28912 df-ch 28926 df-oc 28957 df-ch0 28958 df-shs 29013 df-chj 29015 df-cm 29288 |
This theorem is referenced by: cmcmi 29297 |
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