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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 31838 | . . . . . . 7 ⊢ (𝐶 ∨ℋ 𝐷) ∈ Cℋ |
| 4 | mdslmd.2 | . . . . . . 7 ⊢ 𝐵 ∈ Cℋ | |
| 5 | mdslmd.1 | . . . . . . 7 ⊢ 𝐴 ∈ Cℋ | |
| 6 | 3, 4, 5 | chlej1i 31854 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐷) ∨ℋ 𝐴) ⊆ (𝐵 ∨ℋ 𝐴)) |
| 7 | 1, 2, 5 | chjjdiri 31905 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ∨ℋ 𝐴) = ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) |
| 8 | 4, 5 | chjcomi 31849 | . . . . . 6 ⊢ (𝐵 ∨ℋ 𝐴) = (𝐴 ∨ℋ 𝐵) |
| 9 | 6, 7, 8 | 3sstr3g 3990 | . . . . 5 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵)) |
| 10 | 9 | adantl 487 | . . . 4 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∨ℋ (𝐷 ∨ℋ 𝐴)) ⊆ (𝐴 ∨ℋ 𝐵)) |
| 11 | 5, 1 | chub2i 31851 | . . . . 5 ⊢ 𝐴 ⊆ (𝐶 ∨ℋ 𝐴) |
| 12 | 5, 2 | chub2i 31851 | . . . . 5 ⊢ 𝐴 ⊆ (𝐷 ∨ℋ 𝐴) |
| 13 | 11, 12 | ssini 4192 | . . . 4 ⊢ 𝐴 ⊆ ((𝐶 ∨ℋ 𝐴) ∩ (𝐷 ∨ℋ 𝐴)) |
| 14 | 10, 13 | jctil 529 | . . 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 31838 | . . . 4 ⊢ (𝐶 ∨ℋ 𝐴) ∈ Cℋ |
| 16 | 2, 5 | chjcli 31838 | . . . 4 ⊢ (𝐷 ∨ℋ 𝐴) ∈ Cℋ |
| 17 | 5, 4, 15, 16 | mdslmd1i 32710 | . . 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 605 | . 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 23 | . . . . . 6 ⊢ (𝐴 𝑀ℋ 𝐵 → 𝐴 𝑀ℋ 𝐵) | |
| 20 | inss1 4189 | . . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐶 | |
| 21 | sstr 3946 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐶) → (𝐴 ∩ 𝐵) ⊆ 𝐶) | |
| 22 | 20, 21 | mpan2 704 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐶) |
| 23 | 1, 2 | chub1i 31850 | . . . . . . 7 ⊢ 𝐶 ⊆ (𝐶 ∨ℋ 𝐷) |
| 24 | sstr 3946 | . . . . . . 7 ⊢ ((𝐶 ⊆ (𝐶 ∨ℋ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → 𝐶 ⊆ 𝐵) | |
| 25 | 23, 24 | mpan 703 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → 𝐶 ⊆ 𝐵) |
| 26 | 5, 4, 1 | 3pm3.2i 1358 | . . . . . . 7 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) |
| 27 | mdsl3 32697 | . . . . . . 7 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵)) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) | |
| 28 | 26, 27 | mpan 703 | . . . . . 6 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) |
| 29 | 19, 22, 25, 28 | syl3an 1178 | . . . . 5 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐶 ∨ℋ 𝐴) ∩ 𝐵) = 𝐶) |
| 30 | inss2 4190 | . . . . . . 7 ⊢ (𝐶 ∩ 𝐷) ⊆ 𝐷 | |
| 31 | sstr 3946 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∩ 𝐷) ⊆ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷) | |
| 32 | 30, 31 | mpan2 704 | . . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) → (𝐴 ∩ 𝐵) ⊆ 𝐷) |
| 33 | 2, 1 | chub2i 31851 | . . . . . . 7 ⊢ 𝐷 ⊆ (𝐶 ∨ℋ 𝐷) |
| 34 | sstr 3946 | . . . . . . 7 ⊢ ((𝐷 ⊆ (𝐶 ∨ℋ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → 𝐷 ⊆ 𝐵) | |
| 35 | 33, 34 | mpan 703 | . . . . . 6 ⊢ ((𝐶 ∨ℋ 𝐷) ⊆ 𝐵 → 𝐷 ⊆ 𝐵) |
| 36 | 5, 4, 2 | 3pm3.2i 1358 | . . . . . . 7 ⊢ (𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐷 ∈ Cℋ ) |
| 37 | mdsl3 32697 | . . . . . . 7 ⊢ (((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐷 ∈ Cℋ ) ∧ (𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵)) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) | |
| 38 | 36, 37 | mpan 703 | . . . . . 6 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) |
| 39 | 19, 32, 35, 38 | syl3an 1178 | . . . . 5 ⊢ ((𝐴 𝑀ℋ 𝐵 ∧ (𝐴 ∩ 𝐵) ⊆ (𝐶 ∩ 𝐷) ∧ (𝐶 ∨ℋ 𝐷) ⊆ 𝐵) → ((𝐷 ∨ℋ 𝐴) ∩ 𝐵) = 𝐷) |
| 40 | 29, 39 | breq12d 5124 | . . . 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 1138 | . . 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 728 | . 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 283 | 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 |
| This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∩ cin 3905 ⊆ wss 3906 class class class wbr 5111 (class class class)co 7416 Cℋ cch 31310 ∨ℋ chj 31314 𝑀ℋ cmd 31347 𝑀ℋ* cdmd 31348 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-inf2 9613 ax-cc 10430 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188 ax-pre-sup 11189 ax-addf 11190 ax-mulf 11191 ax-hilex 31380 ax-hfvadd 31381 ax-hvcom 31382 ax-hvass 31383 ax-hv0cl 31384 ax-hvaddid 31385 ax-hfvmul 31386 ax-hvmulid 31387 ax-hvmulass 31388 ax-hvdistr1 31389 ax-hvdistr2 31390 ax-hvmul0 31391 ax-hfi 31460 ax-his1 31463 ax-his2 31464 ax-his3 31465 ax-his4 31466 ax-hcompl 31583 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9325 df-fi 9374 df-sup 9405 df-inf 9406 df-oi 9475 df-card 9937 df-acn 9940 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-div 11883 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12723 df-uz 12874 df-q 12984 df-rp 13028 df-xneg 13148 df-xadd 13149 df-xmul 13150 df-ioo 13387 df-ico 13389 df-icc 13390 df-fz 13547 df-fzo 13695 df-fl 13838 df-seq 14051 df-exp 14111 df-hash 14380 df-cj 15169 df-re 15170 df-im 15171 df-sqrt 15305 df-abs 15306 df-clim 15558 df-rlim 15559 df-sum 15757 df-struct 17224 df-sets 17241 df-slot 17259 df-ndx 17271 df-base 17287 df-ress 17308 df-plusg 17340 df-mulr 17341 df-starv 17342 df-sca 17343 df-vsca 17344 df-ip 17345 df-tset 17346 df-ple 17347 df-ds 17349 df-unif 17350 df-hom 17351 df-cco 17352 df-rest 17492 df-topn 17493 df-0g 17511 df-gsum 17512 df-topgen 17513 df-pt 17514 df-prds 17517 df-xrs 17573 df-qtop 17578 df-imas 17579 df-xps 17581 df-mre 17655 df-mrc 17656 df-acs 17658 df-mgm 18715 df-sgrp 18798 df-mnd 18814 df-submnd 18865 df-mulg 19157 df-cntz 19410 df-cmn 19875 df-psmet 21543 df-xmet 21544 df-met 21545 df-bl 21546 df-mopn 21547 df-fbas 21548 df-fg 21549 df-cnfld 21552 df-top 23080 df-topon 23097 df-topsp 23119 df-bases 23132 df-cld 23205 df-ntr 23206 df-cls 23207 df-nei 23284 df-cn 23413 df-cnp 23414 df-lm 23415 df-haus 23501 df-tx 23748 df-hmeo 23941 df-fil 24032 df-fm 24124 df-flim 24125 df-flf 24126 df-xms 24506 df-ms 24507 df-tms 24508 df-cfil 25443 df-cau 25444 df-cmet 25445 df-grpo 30874 df-gid 30875 df-ginv 30876 df-gdiv 30877 df-ablo 30926 df-vc 30940 df-nv 30973 df-va 30976 df-ba 30977 df-sm 30978 df-0v 30979 df-vs 30980 df-nmcv 30981 df-ims 30982 df-dip 31082 df-ssp 31103 df-ph 31194 df-cbn 31244 df-hnorm 31349 df-hba 31350 df-hvsub 31352 df-hlim 31353 df-hcau 31354 df-sh 31588 df-ch 31602 df-oc 31633 df-ch0 31634 df-shs 31689 df-chj 31691 df-md 32661 df-dmd 32662 |
| This theorem is used by: mdsldmd1i 32712 |
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