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Mirrors > Home > HSE Home > Th. List > dmdbr6ati | Structured version Visualization version GIF version |
Description: Dual modular pair property in terms of atoms. The modular law takes the form of the shearing identity. (Contributed by NM, 18-Jan-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sumdmdi.1 | ⊢ 𝐴 ∈ Cℋ |
sumdmdi.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
dmdbr6ati | ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ HAtoms ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumdmdi.1 | . . . . 5 ⊢ 𝐴 ∈ Cℋ | |
2 | sumdmdi.2 | . . . . 5 ⊢ 𝐵 ∈ Cℋ | |
3 | dmdbr3 30391 | . . . . 5 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) = ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)))) | |
4 | 1, 2, 3 | mp2an 692 | . . . 4 ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) = ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵))) |
5 | chabs2 29603 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝑥 ∩ (𝑥 ∨ℋ 𝐵)) = 𝑥) | |
6 | 2, 5 | mpan2 691 | . . . . . . . . 9 ⊢ (𝑥 ∈ Cℋ → (𝑥 ∩ (𝑥 ∨ℋ 𝐵)) = 𝑥) |
7 | 6 | ineq2d 4132 | . . . . . . . 8 ⊢ (𝑥 ∈ Cℋ → ((𝐴 ∨ℋ 𝐵) ∩ (𝑥 ∩ (𝑥 ∨ℋ 𝐵))) = ((𝐴 ∨ℋ 𝐵) ∩ 𝑥)) |
8 | incom 4120 | . . . . . . . . 9 ⊢ ((𝐴 ∨ℋ 𝐵) ∩ (𝑥 ∩ (𝑥 ∨ℋ 𝐵))) = ((𝑥 ∩ (𝑥 ∨ℋ 𝐵)) ∩ (𝐴 ∨ℋ 𝐵)) | |
9 | inass 4139 | . . . . . . . . 9 ⊢ ((𝑥 ∩ (𝑥 ∨ℋ 𝐵)) ∩ (𝐴 ∨ℋ 𝐵)) = (𝑥 ∩ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵))) | |
10 | incom 4120 | . . . . . . . . 9 ⊢ (𝑥 ∩ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵))) = (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ∩ 𝑥) | |
11 | 8, 9, 10 | 3eqtri 2769 | . . . . . . . 8 ⊢ ((𝐴 ∨ℋ 𝐵) ∩ (𝑥 ∩ (𝑥 ∨ℋ 𝐵))) = (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ∩ 𝑥) |
12 | 7, 11 | eqtr3di 2793 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ∩ 𝑥)) |
13 | 12 | adantr 484 | . . . . . 6 ⊢ ((𝑥 ∈ Cℋ ∧ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) = ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵))) → ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ∩ 𝑥)) |
14 | ineq1 4125 | . . . . . . 7 ⊢ ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) = ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) → ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) = (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ∩ 𝑥)) | |
15 | 14 | adantl 485 | . . . . . 6 ⊢ ((𝑥 ∈ Cℋ ∧ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) = ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵))) → ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) = (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ∩ 𝑥)) |
16 | 13, 15 | eqtr4d 2780 | . . . . 5 ⊢ ((𝑥 ∈ Cℋ ∧ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) = ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵))) → ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥)) |
17 | 16 | ralimiaa 3082 | . . . 4 ⊢ (∀𝑥 ∈ Cℋ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) = ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) → ∀𝑥 ∈ Cℋ ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥)) |
18 | 4, 17 | sylbi 220 | . . 3 ⊢ (𝐴 𝑀ℋ* 𝐵 → ∀𝑥 ∈ Cℋ ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥)) |
19 | atelch 30430 | . . . . 5 ⊢ (𝑥 ∈ HAtoms → 𝑥 ∈ Cℋ ) | |
20 | 19 | imim1i 63 | . . . 4 ⊢ ((𝑥 ∈ Cℋ → ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥)) → (𝑥 ∈ HAtoms → ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥))) |
21 | 20 | ralimi2 3080 | . . 3 ⊢ (∀𝑥 ∈ Cℋ ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) → ∀𝑥 ∈ HAtoms ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥)) |
22 | 18, 21 | syl 17 | . 2 ⊢ (𝐴 𝑀ℋ* 𝐵 → ∀𝑥 ∈ HAtoms ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥)) |
23 | inss1 4148 | . . . . . 6 ⊢ ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) | |
24 | sseq1 3931 | . . . . . 6 ⊢ (((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) → (((𝐴 ∨ℋ 𝐵) ∩ 𝑥) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ↔ ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) | |
25 | 23, 24 | mpbiri 261 | . . . . 5 ⊢ (((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) → ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)) |
26 | incom 4120 | . . . . . . 7 ⊢ ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) | |
27 | df-ss 3888 | . . . . . . . 8 ⊢ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) ↔ (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) = 𝑥) | |
28 | 27 | biimpi 219 | . . . . . . 7 ⊢ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) = 𝑥) |
29 | 26, 28 | syl5eq 2790 | . . . . . 6 ⊢ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) → ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = 𝑥) |
30 | 29 | sseq1d 3937 | . . . . 5 ⊢ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) → (((𝐴 ∨ℋ 𝐵) ∩ 𝑥) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ↔ 𝑥 ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
31 | 25, 30 | syl5ibcom 248 | . . . 4 ⊢ (((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) → (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
32 | 31 | ralimi 3083 | . . 3 ⊢ (∀𝑥 ∈ HAtoms ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) → ∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
33 | 1, 2 | dmdbr5ati 30508 | . . 3 ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ HAtoms (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) → 𝑥 ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
34 | 32, 33 | sylibr 237 | . 2 ⊢ (∀𝑥 ∈ HAtoms ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥) → 𝐴 𝑀ℋ* 𝐵) |
35 | 22, 34 | impbii 212 | 1 ⊢ (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ HAtoms ((𝐴 ∨ℋ 𝐵) ∩ 𝑥) = ((((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ∩ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ∩ cin 3870 ⊆ wss 3871 class class class wbr 5058 (class class class)co 7218 Cℋ cch 29015 ∨ℋ chj 29019 HAtomscat 29051 𝑀ℋ* cdmd 29053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-inf2 9261 ax-cc 10054 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 ax-pre-sup 10812 ax-addf 10813 ax-mulf 10814 ax-hilex 29085 ax-hfvadd 29086 ax-hvcom 29087 ax-hvass 29088 ax-hv0cl 29089 ax-hvaddid 29090 ax-hfvmul 29091 ax-hvmulid 29092 ax-hvmulass 29093 ax-hvdistr1 29094 ax-hvdistr2 29095 ax-hvmul0 29096 ax-hfi 29165 ax-his1 29168 ax-his2 29169 ax-his3 29170 ax-his4 29171 ax-hcompl 29288 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-int 4865 df-iun 4911 df-iin 4912 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-se 5515 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-isom 6394 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-of 7474 df-om 7650 df-1st 7766 df-2nd 7767 df-supp 7909 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-1o 8207 df-2o 8208 df-oadd 8211 df-omul 8212 df-er 8396 df-map 8515 df-pm 8516 df-ixp 8584 df-en 8632 df-dom 8633 df-sdom 8634 df-fin 8635 df-fsupp 8991 df-fi 9032 df-sup 9063 df-inf 9064 df-oi 9131 df-card 9560 df-acn 9563 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-div 11495 df-nn 11836 df-2 11898 df-3 11899 df-4 11900 df-5 11901 df-6 11902 df-7 11903 df-8 11904 df-9 11905 df-n0 12096 df-z 12182 df-dec 12299 df-uz 12444 df-q 12550 df-rp 12592 df-xneg 12709 df-xadd 12710 df-xmul 12711 df-ioo 12944 df-ico 12946 df-icc 12947 df-fz 13101 df-fzo 13244 df-fl 13372 df-seq 13580 df-exp 13641 df-hash 13902 df-cj 14667 df-re 14668 df-im 14669 df-sqrt 14803 df-abs 14804 df-clim 15054 df-rlim 15055 df-sum 15255 df-struct 16705 df-sets 16722 df-slot 16740 df-ndx 16750 df-base 16766 df-ress 16790 df-plusg 16820 df-mulr 16821 df-starv 16822 df-sca 16823 df-vsca 16824 df-ip 16825 df-tset 16826 df-ple 16827 df-ds 16829 df-unif 16830 df-hom 16831 df-cco 16832 df-rest 16932 df-topn 16933 df-0g 16951 df-gsum 16952 df-topgen 16953 df-pt 16954 df-prds 16957 df-xrs 17012 df-qtop 17017 df-imas 17018 df-xps 17020 df-mre 17094 df-mrc 17095 df-acs 17097 df-mgm 18119 df-sgrp 18168 df-mnd 18179 df-submnd 18224 df-mulg 18494 df-cntz 18716 df-cmn 19177 df-psmet 20360 df-xmet 20361 df-met 20362 df-bl 20363 df-mopn 20364 df-fbas 20365 df-fg 20366 df-cnfld 20369 df-top 21796 df-topon 21813 df-topsp 21835 df-bases 21848 df-cld 21921 df-ntr 21922 df-cls 21923 df-nei 22000 df-cn 22129 df-cnp 22130 df-lm 22131 df-haus 22217 df-tx 22464 df-hmeo 22657 df-fil 22748 df-fm 22840 df-flim 22841 df-flf 22842 df-xms 23223 df-ms 23224 df-tms 23225 df-cfil 24157 df-cau 24158 df-cmet 24159 df-grpo 28579 df-gid 28580 df-ginv 28581 df-gdiv 28582 df-ablo 28631 df-vc 28645 df-nv 28678 df-va 28681 df-ba 28682 df-sm 28683 df-0v 28684 df-vs 28685 df-nmcv 28686 df-ims 28687 df-dip 28787 df-ssp 28808 df-ph 28899 df-cbn 28949 df-hnorm 29054 df-hba 29055 df-hvsub 29057 df-hlim 29058 df-hcau 29059 df-sh 29293 df-ch 29307 df-oc 29338 df-ch0 29339 df-shs 29394 df-span 29395 df-chj 29396 df-chsup 29397 df-pjh 29481 df-cv 30365 df-md 30366 df-dmd 30367 df-at 30424 |
This theorem is referenced by: dmdbr7ati 30510 |
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