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Mirrors > Home > HSE Home > Th. List > mdslmd2i | Structured version Visualization version GIF version |
Description: Preservation of the modular pair property in the one-to-one onto mapping between the two sublattices in Lemma 1.3 of [MaedaMaeda] p. 2 (join version). (Contributed by NM, 29-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mdslmd.1 | ⊢ 𝐴 ∈ Cℋ |
mdslmd.2 | ⊢ 𝐵 ∈ Cℋ |
mdslmd.3 | ⊢ 𝐶 ∈ Cℋ |
mdslmd.4 | ⊢ 𝐷 ∈ Cℋ |
Ref | Expression |
---|---|
mdslmd2i | ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → (𝐶 𝑀ℋ 𝐷 ↔ (𝐶 ∨ℋ 𝐴) 𝑀ℋ (𝐷 ∨ℋ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdslmd.3 | . . . . . . . 8 ⊢ 𝐶 ∈ Cℋ | |
2 | mdslmd.4 | . . . . . . . 8 ⊢ 𝐷 ∈ Cℋ | |
3 | 1, 2 | chjcli 31134 | . . . . . . 7 ⊢ (𝐶 ∨ℋ 𝐷) ∈ Cℋ |
4 | mdslmd.2 | . . . . . . 7 ⊢ 𝐵 ∈ Cℋ | |
5 | mdslmd.1 | . . . . . . 7 ⊢ 𝐴 ∈ Cℋ | |
6 | 3, 4, 5 | chlej1i 31150 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐷) ∨ℋ 𝐴) ⊆ (𝐵 ∨ℋ 𝐴)) |
7 | 1, 2, 5 | chjjdiri 31201 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ∨ℋ 𝐴) = ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) |
8 | 4, 5 | chjcomi 31145 | . . . . . 6 ⊢ (𝐵 ∨ℋ 𝐴) = (𝐴 ∨ℋ 𝐵) |
9 | 6, 7, 8 | 3sstr3g 4018 | . . . . 5 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵)) |
10 | 9 | adantl 481 | . . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵)) |
11 | 5, 1 | chub2i 31147 | . . . . 5 ⊢ 𝐴 ⊆ (𝐶 ∨ℋ 𝐴) |
12 | 5, 2 | chub2i 31147 | . . . . 5 ⊢ 𝐴 ⊆ (𝐷 ∨ℋ 𝐴) |
13 | 11, 12 | ssini 4223 | . . . 4 ⊢ 𝐴 ⊆ ((𝐶 ∨ℋ 𝐴) ∩ (𝐷 ∨ℋ 𝐴)) |
14 | 10, 13 | jctil 519 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → (𝐴 ⊆ ((𝐶 ∨ℋ 𝐴) ∩ (𝐷 ∨ℋ 𝐴)) ∧ ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵))) |
15 | 1, 5 | chjcli 31134 | . . . 4 ⊢ (𝐶 ∨ℋ 𝐴) ∈ Cℋ |
16 | 2, 5 | chjcli 31134 | . . . 4 ⊢ (𝐷 ∨ℋ 𝐴) ∈ Cℋ |
17 | 5, 4, 15, 16 | mdslmd1i 32006 | . . 3 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ (𝐴 ⊆ ((𝐶 ∨ℋ 𝐴) ∩ (𝐷 ∨ℋ 𝐴)) ∧ ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵))) → ((𝐶 ∨ℋ 𝐴) 𝑀ℋ (𝐷 ∨ℋ 𝐴) ↔ ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵))) |
18 | 14, 17 | sylan2 592 | . 2 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) 𝑀ℋ (𝐷 ∨ℋ 𝐴) ↔ ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵))) |
19 | id 22 | . . . . . 6 ⊢ (𝐴 𝑀ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵) | |
20 | inss1 4220 | . . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐶 | |
21 | sstr 3982 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
22 | 20, 21 | mpan2 688 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
23 | 1, 2 | chub1i 31146 | . . . . . . 7 ⊢ 𝐶 ⊆ (𝐶 ∨ℋ 𝐷) |
24 | sstr 3982 | . . . . . . 7 ⊢ ((𝐶 ⊆ (𝐶 ∨ℋ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → 𝐶 ⊆ 𝐵) | |
25 | 23, 24 | mpan 687 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → 𝐶 ⊆ 𝐵) |
26 | 5, 4, 1 | 3pm3.2i 1336 | . . . . . . 7 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) |
27 | mdsl3 31993 | . . . . . . 7 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) | |
28 | 26, 27 | mpan 687 | . . . . . 6 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) |
29 | 19, 22, 25, 28 | syl3an 1157 | . . . . 5 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) |
30 | inss2 4221 | . . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐷 | |
31 | sstr 3982 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷) | |
32 | 30, 31 | mpan2 688 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷) |
33 | 2, 1 | chub2i 31147 | . . . . . . 7 ⊢ 𝐷 ⊆ (𝐶 ∨ℋ 𝐷) |
34 | sstr 3982 | . . . . . . 7 ⊢ ((𝐷 ⊆ (𝐶 ∨ℋ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → 𝐷 ⊆ 𝐵) | |
35 | 33, 34 | mpan 687 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → 𝐷 ⊆ 𝐵) |
36 | 5, 4, 2 | 3pm3.2i 1336 | . . . . . . 7 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐷 ∈ Cℋ ) |
37 | mdsl3 31993 | . . . . . . 7 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐷 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵)) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) | |
38 | 36, 37 | mpan 687 | . . . . . 6 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) |
39 | 19, 32, 35, 38 | syl3an 1157 | . . . . 5 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) |
40 | 29, 39 | breq12d 5151 | . . . 4 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → (((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) ↔ 𝐶 𝑀ℋ 𝐷)) |
41 | 40 | 3expb 1117 | . . 3 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → (((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) ↔ 𝐶 𝑀ℋ 𝐷)) |
42 | 41 | adantlr 712 | . 2 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → (((𝐶 ∨ℋ 𝐴) ∩ 𝐵) 𝑀ℋ ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) ↔ 𝐶 𝑀ℋ 𝐷)) |
43 | 18, 42 | bitr2d 280 | 1 ⊢ (((𝐴 𝑀ℋ 𝐵 ∧ 𝐵 𝑀ℋ* 𝐴) ∧ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵)) → (𝐶 𝑀ℋ 𝐷 ↔ (𝐶 ∨ℋ 𝐴) 𝑀ℋ (𝐷 ∨ℋ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3939 ⊆ wss 3940 class class class wbr 5138 (class class class)co 7401 Cℋ cch 30606 ∨ℋ chj 30610 𝑀ℋ cmd 30643 𝑀ℋ* cdmd 30644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-inf2 9631 ax-cc 10425 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 ax-addf 11184 ax-mulf 11185 ax-hilex 30676 ax-hfvadd 30677 ax-hvcom 30678 ax-hvass 30679 ax-hv0cl 30680 ax-hvaddid 30681 ax-hfvmul 30682 ax-hvmulid 30683 ax-hvmulass 30684 ax-hvdistr1 30685 ax-hvdistr2 30686 ax-hvmul0 30687 ax-hfi 30756 ax-his1 30759 ax-his2 30760 ax-his3 30761 ax-his4 30762 ax-hcompl 30879 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-se 5622 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-oadd 8465 df-omul 8466 df-er 8698 df-map 8817 df-pm 8818 df-ixp 8887 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-fi 9401 df-sup 9432 df-inf 9433 df-oi 9500 df-card 9929 df-acn 9932 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18560 df-sgrp 18639 df-mnd 18655 df-submnd 18701 df-mulg 18983 df-cntz 19218 df-cmn 19687 df-psmet 21215 df-xmet 21216 df-met 21217 df-bl 21218 df-mopn 21219 df-fbas 21220 df-fg 21221 df-cnfld 21224 df-top 22706 df-topon 22723 df-topsp 22745 df-bases 22759 df-cld 22833 df-ntr 22834 df-cls 22835 df-nei 22912 df-cn 23041 df-cnp 23042 df-lm 23043 df-haus 23129 df-tx 23376 df-hmeo 23569 df-fil 23660 df-fm 23752 df-flim 23753 df-flf 23754 df-xms 24136 df-ms 24137 df-tms 24138 df-cfil 25093 df-cau 25094 df-cmet 25095 df-grpo 30170 df-gid 30171 df-ginv 30172 df-gdiv 30173 df-ablo 30222 df-vc 30236 df-nv 30269 df-va 30272 df-ba 30273 df-sm 30274 df-0v 30275 df-vs 30276 df-nmcv 30277 df-ims 30278 df-dip 30378 df-ssp 30399 df-ph 30490 df-cbn 30540 df-hnorm 30645 df-hba 30646 df-hvsub 30648 df-hlim 30649 df-hcau 30650 df-sh 30884 df-ch 30898 df-oc 30929 df-ch0 30930 df-shs 30985 df-chj 30987 df-md 31957 df-dmd 31958 |
This theorem is referenced by: mdsldmd1i 32008 |
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