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Mirrors > Home > HSE Home > Th. List > mdsl2i | Structured version Visualization version GIF version |
Description: If the modular pair property holds in a sublattice, it holds in the whole lattice. Lemma 1.4 of [MaedaMaeda] p. 2. (Contributed by NM, 28-Apr-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mdsl.1 | ⊢ 𝐴 ∈ Cℋ |
mdsl.2 | ⊢ 𝐵 ∈ Cℋ |
Ref | Expression |
---|---|
mdsl2i | ⊢ (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mdsl.1 | . . . . . . . . . . . 12 ⊢ 𝐴 ∈ Cℋ | |
2 | chub1 30396 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → 𝑥 ⊆ (𝑥 ∨ℋ 𝐴)) | |
3 | 1, 2 | mpan2 689 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ Cℋ → 𝑥 ⊆ (𝑥 ∨ℋ 𝐴)) |
4 | iba 528 | . . . . . . . . . . . 12 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ (𝑥 ∨ℋ 𝐴) ↔ (𝑥 ⊆ (𝑥 ∨ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵))) | |
5 | ssin 4190 | . . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ (𝑥 ∨ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)) | |
6 | 4, 5 | bitrdi 286 | . . . . . . . . . . 11 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ (𝑥 ∨ℋ 𝐴) ↔ 𝑥 ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))) |
7 | 3, 6 | syl5ibcom 244 | . . . . . . . . . 10 ⊢ (𝑥 ∈ Cℋ → (𝑥 ⊆ 𝐵 → 𝑥 ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))) |
8 | chub2 30397 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → 𝐴 ⊆ (𝑥 ∨ℋ 𝐴)) | |
9 | 1, 8 | mpan 688 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ Cℋ → 𝐴 ⊆ (𝑥 ∨ℋ 𝐴)) |
10 | 9 | ssrind 4195 | . . . . . . . . . 10 ⊢ (𝑥 ∈ Cℋ → (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)) |
11 | 7, 10 | jctird 527 | . . . . . . . . 9 ⊢ (𝑥 ∈ Cℋ → (𝑥 ⊆ 𝐵 → (𝑥 ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)))) |
12 | chjcl 30246 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝑥 ∨ℋ 𝐴) ∈ Cℋ ) | |
13 | 1, 12 | mpan2 689 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ Cℋ → (𝑥 ∨ℋ 𝐴) ∈ Cℋ ) |
14 | mdsl.2 | . . . . . . . . . . . 12 ⊢ 𝐵 ∈ Cℋ | |
15 | chincl 30388 | . . . . . . . . . . . 12 ⊢ (((𝑥 ∨ℋ 𝐴) ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∈ Cℋ ) | |
16 | 14, 15 | mpan2 689 | . . . . . . . . . . 11 ⊢ ((𝑥 ∨ℋ 𝐴) ∈ Cℋ → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∈ Cℋ ) |
17 | 13, 16 | syl 17 | . . . . . . . . . 10 ⊢ (𝑥 ∈ Cℋ → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∈ Cℋ ) |
18 | 1, 14 | chincli 30349 | . . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
19 | chlub 30398 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ Cℋ ∧ (𝐴 ∩ 𝐵) ∈ Cℋ ∧ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∈ Cℋ ) → ((𝑥 ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)) ↔ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))) | |
20 | 18, 19 | mp3an2 1449 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ Cℋ ∧ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∈ Cℋ ) → ((𝑥 ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)) ↔ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))) |
21 | 17, 20 | mpdan 685 | . . . . . . . . 9 ⊢ (𝑥 ∈ Cℋ → ((𝑥 ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)) ↔ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))) |
22 | 11, 21 | sylibd 238 | . . . . . . . 8 ⊢ (𝑥 ∈ Cℋ → (𝑥 ⊆ 𝐵 → (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))) |
23 | eqss 3959 | . . . . . . . . 9 ⊢ (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ∧ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵))) | |
24 | 23 | rbaib 539 | . . . . . . . 8 ⊢ ((𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) |
25 | 22, 24 | syl6 35 | . . . . . . 7 ⊢ (𝑥 ∈ Cℋ → (𝑥 ⊆ 𝐵 → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))) |
26 | 25 | adantld 491 | . . . . . 6 ⊢ (𝑥 ∈ Cℋ → (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → (((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))) |
27 | 26 | pm5.74d 272 | . . . . 5 ⊢ (𝑥 ∈ Cℋ → ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))) |
28 | 14, 1 | chub2i 30359 | . . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐴 ∨ℋ 𝐵) |
29 | sstr 3952 | . . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐴 ∨ℋ 𝐵)) → 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) | |
30 | 28, 29 | mpan2 689 | . . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) |
31 | 30 | pm4.71ri 561 | . . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 ↔ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵)) |
32 | 31 | anbi2i 623 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵))) |
33 | anass 469 | . . . . . . 7 ⊢ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵))) | |
34 | 32, 33 | bitr4i 277 | . . . . . 6 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵)) |
35 | 34 | imbi1i 349 | . . . . 5 ⊢ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) |
36 | 27, 35 | bitr3di 285 | . . . 4 ⊢ (𝑥 ∈ Cℋ → ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))) |
37 | impexp 451 | . . . 4 ⊢ (((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))) | |
38 | 36, 37 | bitrdi 286 | . . 3 ⊢ (𝑥 ∈ Cℋ → ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))))) |
39 | 38 | ralbiia 3094 | . 2 ⊢ (∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵))) ↔ ∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵))))) |
40 | 1, 14 | mdsl1i 31210 | . 2 ⊢ (∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) → (𝑥 ⊆ 𝐵 → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) = (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) ↔ 𝐴 𝑀ℋ 𝐵) |
41 | 39, 40 | bitr2i 275 | 1 ⊢ (𝐴 𝑀ℋ 𝐵 ↔ ∀𝑥 ∈ Cℋ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ⊆ (𝑥 ∨ℋ (𝐴 ∩ 𝐵)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ∩ cin 3909 ⊆ wss 3910 class class class wbr 5105 (class class class)co 7356 Cℋ cch 29818 ∨ℋ chj 29822 𝑀ℋ cmd 29855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-inf2 9576 ax-cc 10370 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-pre-sup 11128 ax-addf 11129 ax-mulf 11130 ax-hilex 29888 ax-hfvadd 29889 ax-hvcom 29890 ax-hvass 29891 ax-hv0cl 29892 ax-hvaddid 29893 ax-hfvmul 29894 ax-hvmulid 29895 ax-hvmulass 29896 ax-hvdistr1 29897 ax-hvdistr2 29898 ax-hvmul0 29899 ax-hfi 29968 ax-his1 29971 ax-his2 29972 ax-his3 29973 ax-his4 29974 ax-hcompl 30091 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-supp 8092 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-omul 8416 df-er 8647 df-map 8766 df-pm 8767 df-ixp 8835 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-fsupp 9305 df-fi 9346 df-sup 9377 df-inf 9378 df-oi 9445 df-card 9874 df-acn 9877 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-7 12220 df-8 12221 df-9 12222 df-n0 12413 df-z 12499 df-dec 12618 df-uz 12763 df-q 12873 df-rp 12915 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-ioo 13267 df-ico 13269 df-icc 13270 df-fz 13424 df-fzo 13567 df-fl 13696 df-seq 13906 df-exp 13967 df-hash 14230 df-cj 14983 df-re 14984 df-im 14985 df-sqrt 15119 df-abs 15120 df-clim 15369 df-rlim 15370 df-sum 15570 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-starv 17147 df-sca 17148 df-vsca 17149 df-ip 17150 df-tset 17151 df-ple 17152 df-ds 17154 df-unif 17155 df-hom 17156 df-cco 17157 df-rest 17303 df-topn 17304 df-0g 17322 df-gsum 17323 df-topgen 17324 df-pt 17325 df-prds 17328 df-xrs 17383 df-qtop 17388 df-imas 17389 df-xps 17391 df-mre 17465 df-mrc 17466 df-acs 17468 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-submnd 18601 df-mulg 18871 df-cntz 19095 df-cmn 19562 df-psmet 20786 df-xmet 20787 df-met 20788 df-bl 20789 df-mopn 20790 df-fbas 20791 df-fg 20792 df-cnfld 20795 df-top 22241 df-topon 22258 df-topsp 22280 df-bases 22294 df-cld 22368 df-ntr 22369 df-cls 22370 df-nei 22447 df-cn 22576 df-cnp 22577 df-lm 22578 df-haus 22664 df-tx 22911 df-hmeo 23104 df-fil 23195 df-fm 23287 df-flim 23288 df-flf 23289 df-xms 23671 df-ms 23672 df-tms 23673 df-cfil 24617 df-cau 24618 df-cmet 24619 df-grpo 29382 df-gid 29383 df-ginv 29384 df-gdiv 29385 df-ablo 29434 df-vc 29448 df-nv 29481 df-va 29484 df-ba 29485 df-sm 29486 df-0v 29487 df-vs 29488 df-nmcv 29489 df-ims 29490 df-dip 29590 df-ssp 29611 df-ph 29702 df-cbn 29752 df-hnorm 29857 df-hba 29858 df-hvsub 29860 df-hlim 29861 df-hcau 29862 df-sh 30096 df-ch 30110 df-oc 30141 df-ch0 30142 df-shs 30197 df-chj 30199 df-md 31169 |
This theorem is referenced by: mdsl2bi 31212 mdslmd1i 31218 |
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