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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 30498 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → 𝑥 ⊆ (𝑥 ∨ℋ 𝐴)) | |
3 | 1, 2 | mpan2 690 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ Cℋ → 𝑥 ⊆ (𝑥 ∨ℋ 𝐴)) |
4 | iba 529 | . . . . . . . . . . . 12 ⊢ (𝑥 ⊆ 𝐵 → (𝑥 ⊆ (𝑥 ∨ℋ 𝐴) ↔ (𝑥 ⊆ (𝑥 ∨ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵))) | |
5 | ssin 4194 | . . . . . . . . . . . 12 ⊢ ((𝑥 ⊆ (𝑥 ∨ℋ 𝐴) ∧ 𝑥 ⊆ 𝐵) ↔ 𝑥 ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)) | |
6 | 4, 5 | bitrdi 287 | . . . . . . . . . . 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 30499 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → 𝐴 ⊆ (𝑥 ∨ℋ 𝐴)) | |
9 | 1, 8 | mpan 689 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ Cℋ → 𝐴 ⊆ (𝑥 ∨ℋ 𝐴)) |
10 | 9 | ssrind 4199 | . . . . . . . . . 10 ⊢ (𝑥 ∈ Cℋ → (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)) |
11 | 7, 10 | jctird 528 | . . . . . . . . 9 ⊢ (𝑥 ∈ Cℋ → (𝑥 ⊆ 𝐵 → (𝑥 ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∧ (𝐴 ∩ 𝐵) ⊆ ((𝑥 ∨ℋ 𝐴) ∩ 𝐵)))) |
12 | chjcl 30348 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐴 ∈ Cℋ ) → (𝑥 ∨ℋ 𝐴) ∈ Cℋ ) | |
13 | 1, 12 | mpan2 690 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ Cℋ → (𝑥 ∨ℋ 𝐴) ∈ Cℋ ) |
14 | mdsl.2 | . . . . . . . . . . . 12 ⊢ 𝐵 ∈ Cℋ | |
15 | chincl 30490 | . . . . . . . . . . . 12 ⊢ (((𝑥 ∨ℋ 𝐴) ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → ((𝑥 ∨ℋ 𝐴) ∩ 𝐵) ∈ Cℋ ) | |
16 | 14, 15 | mpan2 690 | . . . . . . . . . . 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 30451 | . . . . . . . . . . 11 ⊢ (𝐴 ∩ 𝐵) ∈ Cℋ |
19 | chlub 30500 | . . . . . . . . . . 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 1450 | . . . . . . . . . 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 686 | . . . . . . . . 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 3963 | . . . . . . . . 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 540 | . . . . . . . 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 492 | . . . . . 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 273 | . . . . 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 30461 | . . . . . . . . . 10 ⊢ 𝐵 ⊆ (𝐴 ∨ℋ 𝐵) |
29 | sstr 3956 | . . . . . . . . . 10 ⊢ ((𝑥 ⊆ 𝐵 ∧ 𝐵 ⊆ (𝐴 ∨ℋ 𝐵)) → 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) | |
30 | 28, 29 | mpan2 690 | . . . . . . . . 9 ⊢ (𝑥 ⊆ 𝐵 → 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) |
31 | 30 | pm4.71ri 562 | . . . . . . . 8 ⊢ (𝑥 ⊆ 𝐵 ↔ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵)) |
32 | 31 | anbi2i 624 | . . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵))) |
33 | anass 470 | . . . . . . 7 ⊢ ((((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵) ↔ ((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ (𝑥 ⊆ (𝐴 ∨ℋ 𝐵) ∧ 𝑥 ⊆ 𝐵))) | |
34 | 32, 33 | bitr4i 278 | . . . . . 6 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ 𝐵) ↔ (((𝐴 ∩ 𝐵) ⊆ 𝑥 ∧ 𝑥 ⊆ (𝐴 ∨ℋ 𝐵)) ∧ 𝑥 ⊆ 𝐵)) |
35 | 34 | imbi1i 350 | . . . . 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 286 | . . . 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 452 | . . . 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 287 | . . 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 3091 | . 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 31312 | . 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 276 | 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 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ∩ cin 3913 ⊆ wss 3914 class class class wbr 5109 (class class class)co 7361 Cℋ cch 29920 ∨ℋ chj 29924 𝑀ℋ cmd 29957 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cc 10379 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 ax-hilex 29990 ax-hfvadd 29991 ax-hvcom 29992 ax-hvass 29993 ax-hv0cl 29994 ax-hvaddid 29995 ax-hfvmul 29996 ax-hvmulid 29997 ax-hvmulass 29998 ax-hvdistr1 29999 ax-hvdistr2 30000 ax-hvmul0 30001 ax-hfi 30070 ax-his1 30073 ax-his2 30074 ax-his3 30075 ax-his4 30076 ax-hcompl 30193 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-oadd 8420 df-omul 8421 df-er 8654 df-map 8773 df-pm 8774 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-acn 9886 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-rlim 15380 df-sum 15580 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-mulg 18881 df-cntz 19105 df-cmn 19572 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-fbas 20816 df-fg 20817 df-cnfld 20820 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cld 22393 df-ntr 22394 df-cls 22395 df-nei 22472 df-cn 22601 df-cnp 22602 df-lm 22603 df-haus 22689 df-tx 22936 df-hmeo 23129 df-fil 23220 df-fm 23312 df-flim 23313 df-flf 23314 df-xms 23696 df-ms 23697 df-tms 23698 df-cfil 24642 df-cau 24643 df-cmet 24644 df-grpo 29484 df-gid 29485 df-ginv 29486 df-gdiv 29487 df-ablo 29536 df-vc 29550 df-nv 29583 df-va 29586 df-ba 29587 df-sm 29588 df-0v 29589 df-vs 29590 df-nmcv 29591 df-ims 29592 df-dip 29692 df-ssp 29713 df-ph 29804 df-cbn 29854 df-hnorm 29959 df-hba 29960 df-hvsub 29962 df-hlim 29963 df-hcau 29964 df-sh 30198 df-ch 30212 df-oc 30243 df-ch0 30244 df-shs 30299 df-chj 30301 df-md 31271 |
This theorem is referenced by: mdsl2bi 31314 mdslmd1i 31320 |
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