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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 30698 | . . . . . . 7 ⊢ (𝐶 ∨ℋ 𝐷) ∈ Cℋ |
| 4 | mdslmd.2 | . . . . . . 7 ⊢ 𝐵 ∈ Cℋ | |
| 5 | mdslmd.1 | . . . . . . 7 ⊢ 𝐴 ∈ Cℋ | |
| 6 | 3, 4, 5 | chlej1i 30714 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐷) ∨ℋ 𝐴) ⊆ (𝐵 ∨ℋ 𝐴)) |
| 7 | 1, 2, 5 | chjjdiri 30765 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ∨ℋ 𝐴) = ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) |
| 8 | 4, 5 | chjcomi 30709 | . . . . . 6 ⊢ (𝐵 ∨ℋ 𝐴) = (𝐴 ∨ℋ 𝐵) |
| 9 | 6, 7, 8 | 3sstr3g 4026 | . . . . 5 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵)) |
| 10 | 9 | adantl 483 | . . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵)) |
| 11 | 5, 1 | chub2i 30711 | . . . . 5 ⊢ 𝐴 ⊆ (𝐶 ∨ℋ 𝐴) |
| 12 | 5, 2 | chub2i 30711 | . . . . 5 ⊢ 𝐴 ⊆ (𝐷 ∨ℋ 𝐴) |
| 13 | 11, 12 | ssini 4231 | . . . 4 ⊢ 𝐴 ⊆ ((𝐶 ∨ℋ 𝐴) ∩ (𝐷 ∨ℋ 𝐴)) |
| 14 | 10, 13 | jctil 521 | . . 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 30698 | . . . 4 ⊢ (𝐶 ∨ℋ 𝐴) ∈ Cℋ |
| 16 | 2, 5 | chjcli 30698 | . . . 4 ⊢ (𝐷 ∨ℋ 𝐴) ∈ Cℋ |
| 17 | 5, 4, 15, 16 | mdslmd1i 31570 | . . 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 594 | . 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 4228 | . . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐶 | |
| 21 | sstr 3990 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
| 22 | 20, 21 | mpan2 690 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
| 23 | 1, 2 | chub1i 30710 | . . . . . . 7 ⊢ 𝐶 ⊆ (𝐶 ∨ℋ 𝐷) |
| 24 | sstr 3990 | . . . . . . 7 ⊢ ((𝐶 ⊆ (𝐶 ∨ℋ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → 𝐶 ⊆ 𝐵) | |
| 25 | 23, 24 | mpan 689 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → 𝐶 ⊆ 𝐵) |
| 26 | 5, 4, 1 | 3pm3.2i 1340 | . . . . . . 7 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) |
| 27 | mdsl3 31557 | . . . . . . 7 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) | |
| 28 | 26, 27 | mpan 689 | . . . . . 6 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) |
| 29 | 19, 22, 25, 28 | syl3an 1161 | . . . . 5 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) |
| 30 | inss2 4229 | . . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐷 | |
| 31 | sstr 3990 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷) | |
| 32 | 30, 31 | mpan2 690 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷) |
| 33 | 2, 1 | chub2i 30711 | . . . . . . 7 ⊢ 𝐷 ⊆ (𝐶 ∨ℋ 𝐷) |
| 34 | sstr 3990 | . . . . . . 7 ⊢ ((𝐷 ⊆ (𝐶 ∨ℋ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → 𝐷 ⊆ 𝐵) | |
| 35 | 33, 34 | mpan 689 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → 𝐷 ⊆ 𝐵) |
| 36 | 5, 4, 2 | 3pm3.2i 1340 | . . . . . . 7 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐷 ∈ Cℋ ) |
| 37 | mdsl3 31557 | . . . . . . 7 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐷 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵)) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) | |
| 38 | 36, 37 | mpan 689 | . . . . . 6 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) |
| 39 | 19, 32, 35, 38 | syl3an 1161 | . . . . 5 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) |
| 40 | 29, 39 | breq12d 5161 | . . . 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 1121 | . . 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 714 | . 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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∩ cin 3947 ⊆ wss 3948 class class class wbr 5148 (class class class)co 7406 Cℋ cch 30170 ∨ℋ chj 30174 𝑀ℋ cmd 30207 𝑀ℋ* cdmd 30208 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-cc 10427 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 ax-addf 11186 ax-mulf 11187 ax-hilex 30240 ax-hfvadd 30241 ax-hvcom 30242 ax-hvass 30243 ax-hv0cl 30244 ax-hvaddid 30245 ax-hfvmul 30246 ax-hvmulid 30247 ax-hvmulass 30248 ax-hvdistr1 30249 ax-hvdistr2 30250 ax-hvmul0 30251 ax-hfi 30320 ax-his1 30323 ax-his2 30324 ax-his3 30325 ax-his4 30326 ax-hcompl 30443 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-of 7667 df-om 7853 df-1st 7972 df-2nd 7973 df-supp 8144 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-oadd 8467 df-omul 8468 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9502 df-card 9931 df-acn 9934 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-q 12930 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-ioo 13325 df-ico 13327 df-icc 13328 df-fz 13482 df-fzo 13625 df-fl 13754 df-seq 13964 df-exp 14025 df-hash 14288 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-clim 15429 df-rlim 15430 df-sum 15630 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-starv 17209 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-hom 17218 df-cco 17219 df-rest 17365 df-topn 17366 df-0g 17384 df-gsum 17385 df-topgen 17386 df-pt 17387 df-prds 17390 df-xrs 17445 df-qtop 17450 df-imas 17451 df-xps 17453 df-mre 17527 df-mrc 17528 df-acs 17530 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-mulg 18946 df-cntz 19176 df-cmn 19645 df-psmet 20929 df-xmet 20930 df-met 20931 df-bl 20932 df-mopn 20933 df-fbas 20934 df-fg 20935 df-cnfld 20938 df-top 22388 df-topon 22405 df-topsp 22427 df-bases 22441 df-cld 22515 df-ntr 22516 df-cls 22517 df-nei 22594 df-cn 22723 df-cnp 22724 df-lm 22725 df-haus 22811 df-tx 23058 df-hmeo 23251 df-fil 23342 df-fm 23434 df-flim 23435 df-flf 23436 df-xms 23818 df-ms 23819 df-tms 23820 df-cfil 24764 df-cau 24765 df-cmet 24766 df-grpo 29734 df-gid 29735 df-ginv 29736 df-gdiv 29737 df-ablo 29786 df-vc 29800 df-nv 29833 df-va 29836 df-ba 29837 df-sm 29838 df-0v 29839 df-vs 29840 df-nmcv 29841 df-ims 29842 df-dip 29942 df-ssp 29963 df-ph 30054 df-cbn 30104 df-hnorm 30209 df-hba 30210 df-hvsub 30212 df-hlim 30213 df-hcau 30214 df-sh 30448 df-ch 30462 df-oc 30493 df-ch0 30494 df-shs 30549 df-chj 30551 df-md 31521 df-dmd 31522 |
| This theorem is referenced by: mdsldmd1i 31572 |
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