![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > dmdbr4 | Structured version Visualization version GIF version |
Description: Binary relation expressing the dual modular pair property. This version quantifies an ordering instead of an inference. (Contributed by NM, 6-Jul-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dmdbr4 | ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmdbr2 29706 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)))) | |
2 | chub2 28911 | . . . . . . . . 9 ⊢ ((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → 𝐵 ⊆ (𝑥 ∨ℋ 𝐵)) | |
3 | 2 | ancoms 452 | . . . . . . . 8 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → 𝐵 ⊆ (𝑥 ∨ℋ 𝐵)) |
4 | chjcl 28760 | . . . . . . . . 9 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝑥 ∨ℋ 𝐵) ∈ Cℋ ) | |
5 | sseq2 3852 | . . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 ∨ℋ 𝐵) → (𝐵 ⊆ 𝑦 ↔ 𝐵 ⊆ (𝑥 ∨ℋ 𝐵))) | |
6 | ineq1 4034 | . . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 ∨ℋ 𝐵) → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) = ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵))) | |
7 | ineq1 4034 | . . . . . . . . . . . . 13 ⊢ (𝑦 = (𝑥 ∨ℋ 𝐵) → (𝑦 ∩ 𝐴) = ((𝑥 ∨ℋ 𝐵) ∩ 𝐴)) | |
8 | 7 | oveq1d 6920 | . . . . . . . . . . . 12 ⊢ (𝑦 = (𝑥 ∨ℋ 𝐵) → ((𝑦 ∩ 𝐴) ∨ℋ 𝐵) = (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)) |
9 | 6, 8 | sseq12d 3859 | . . . . . . . . . . 11 ⊢ (𝑦 = (𝑥 ∨ℋ 𝐵) → ((𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵) ↔ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
10 | 5, 9 | imbi12d 336 | . . . . . . . . . 10 ⊢ (𝑦 = (𝑥 ∨ℋ 𝐵) → ((𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) ↔ (𝐵 ⊆ (𝑥 ∨ℋ 𝐵) → ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)))) |
11 | 10 | rspcv 3522 | . . . . . . . . 9 ⊢ ((𝑥 ∨ℋ 𝐵) ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) → (𝐵 ⊆ (𝑥 ∨ℋ 𝐵) → ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)))) |
12 | 4, 11 | syl 17 | . . . . . . . 8 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) → (𝐵 ⊆ (𝑥 ∨ℋ 𝐵) → ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)))) |
13 | 3, 12 | mpid 44 | . . . . . . 7 ⊢ ((𝑥 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) → ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
14 | 13 | ex 403 | . . . . . 6 ⊢ (𝑥 ∈ Cℋ → (𝐵 ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) → ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)))) |
15 | 14 | com3l 89 | . . . . 5 ⊢ (𝐵 ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) → (𝑥 ∈ Cℋ → ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵)))) |
16 | 15 | ralrimdv 3177 | . . . 4 ⊢ (𝐵 ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) → ∀𝑥 ∈ Cℋ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
17 | chlejb2 28916 | . . . . . . . . . . . 12 ⊢ ((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → (𝐵 ⊆ 𝑥 ↔ (𝑥 ∨ℋ 𝐵) = 𝑥)) | |
18 | 17 | biimpa 470 | . . . . . . . . . . 11 ⊢ (((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) ∧ 𝐵 ⊆ 𝑥) → (𝑥 ∨ℋ 𝐵) = 𝑥) |
19 | 18 | ineq1d 4040 | . . . . . . . . . 10 ⊢ (((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) ∧ 𝐵 ⊆ 𝑥) → ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) = (𝑥 ∩ (𝐴 ∨ℋ 𝐵))) |
20 | 18 | ineq1d 4040 | . . . . . . . . . . 11 ⊢ (((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) ∧ 𝐵 ⊆ 𝑥) → ((𝑥 ∨ℋ 𝐵) ∩ 𝐴) = (𝑥 ∩ 𝐴)) |
21 | 20 | oveq1d 6920 | . . . . . . . . . 10 ⊢ (((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) ∧ 𝐵 ⊆ 𝑥) → (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) = ((𝑥 ∩ 𝐴) ∨ℋ 𝐵)) |
22 | 19, 21 | sseq12d 3859 | . . . . . . . . 9 ⊢ (((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) ∧ 𝐵 ⊆ 𝑥) → (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) ↔ (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵))) |
23 | 22 | biimpd 221 | . . . . . . . 8 ⊢ (((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) ∧ 𝐵 ⊆ 𝑥) → (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵))) |
24 | 23 | ex 403 | . . . . . . 7 ⊢ ((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → (𝐵 ⊆ 𝑥 → (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵)))) |
25 | 24 | com23 86 | . . . . . 6 ⊢ ((𝐵 ∈ Cℋ ∧ 𝑥 ∈ Cℋ ) → (((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) → (𝐵 ⊆ 𝑥 → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵)))) |
26 | 25 | ralimdva 3171 | . . . . 5 ⊢ (𝐵 ∈ Cℋ → (∀𝑥 ∈ Cℋ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) → ∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵)))) |
27 | sseq2 3852 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐵 ⊆ 𝑥 ↔ 𝐵 ⊆ 𝑦)) | |
28 | ineq1 4034 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) = (𝑦 ∩ (𝐴 ∨ℋ 𝐵))) | |
29 | ineq1 4034 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ∩ 𝐴) = (𝑦 ∩ 𝐴)) | |
30 | 29 | oveq1d 6920 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) = ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) |
31 | 28, 30 | sseq12d 3859 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵) ↔ (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵))) |
32 | 27, 31 | imbi12d 336 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐵 ⊆ 𝑥 → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵)) ↔ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)))) |
33 | 32 | cbvralv 3383 | . . . . 5 ⊢ (∀𝑥 ∈ Cℋ (𝐵 ⊆ 𝑥 → (𝑥 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑥 ∩ 𝐴) ∨ℋ 𝐵)) ↔ ∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵))) |
34 | 26, 33 | syl6ib 243 | . . . 4 ⊢ (𝐵 ∈ Cℋ → (∀𝑥 ∈ Cℋ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵) → ∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)))) |
35 | 16, 34 | impbid 204 | . . 3 ⊢ (𝐵 ∈ Cℋ → (∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) ↔ ∀𝑥 ∈ Cℋ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
36 | 35 | adantl 475 | . 2 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (∀𝑦 ∈ Cℋ (𝐵 ⊆ 𝑦 → (𝑦 ∩ (𝐴 ∨ℋ 𝐵)) ⊆ ((𝑦 ∩ 𝐴) ∨ℋ 𝐵)) ↔ ∀𝑥 ∈ Cℋ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
37 | 1, 36 | bitrd 271 | 1 ⊢ ((𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ) → (𝐴 𝑀ℋ* 𝐵 ↔ ∀𝑥 ∈ Cℋ ((𝑥 ∨ℋ 𝐵) ∩ (𝐴 ∨ℋ 𝐵)) ⊆ (((𝑥 ∨ℋ 𝐵) ∩ 𝐴) ∨ℋ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ∩ cin 3797 ⊆ wss 3798 class class class wbr 4873 (class class class)co 6905 Cℋ cch 28330 ∨ℋ chj 28334 𝑀ℋ* cdmd 28368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-inf2 8815 ax-cc 9572 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 ax-addf 10331 ax-mulf 10332 ax-hilex 28400 ax-hfvadd 28401 ax-hvcom 28402 ax-hvass 28403 ax-hv0cl 28404 ax-hvaddid 28405 ax-hfvmul 28406 ax-hvmulid 28407 ax-hvmulass 28408 ax-hvdistr1 28409 ax-hvdistr2 28410 ax-hvmul0 28411 ax-hfi 28480 ax-his1 28483 ax-his2 28484 ax-his3 28485 ax-his4 28486 ax-hcompl 28603 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-se 5302 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-isom 6132 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-supp 7560 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-2o 7827 df-oadd 7830 df-omul 7831 df-er 8009 df-map 8124 df-pm 8125 df-ixp 8176 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-fsupp 8545 df-fi 8586 df-sup 8617 df-inf 8618 df-oi 8684 df-card 9078 df-acn 9081 df-cda 9305 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-q 12072 df-rp 12113 df-xneg 12232 df-xadd 12233 df-xmul 12234 df-ioo 12467 df-ico 12469 df-icc 12470 df-fz 12620 df-fzo 12761 df-fl 12888 df-seq 13096 df-exp 13155 df-hash 13411 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-clim 14596 df-rlim 14597 df-sum 14794 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-starv 16320 df-sca 16321 df-vsca 16322 df-ip 16323 df-tset 16324 df-ple 16325 df-ds 16327 df-unif 16328 df-hom 16329 df-cco 16330 df-rest 16436 df-topn 16437 df-0g 16455 df-gsum 16456 df-topgen 16457 df-pt 16458 df-prds 16461 df-xrs 16515 df-qtop 16520 df-imas 16521 df-xps 16523 df-mre 16599 df-mrc 16600 df-acs 16602 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-mulg 17895 df-cntz 18100 df-cmn 18548 df-psmet 20098 df-xmet 20099 df-met 20100 df-bl 20101 df-mopn 20102 df-fbas 20103 df-fg 20104 df-cnfld 20107 df-top 21069 df-topon 21086 df-topsp 21108 df-bases 21121 df-cld 21194 df-ntr 21195 df-cls 21196 df-nei 21273 df-cn 21402 df-cnp 21403 df-lm 21404 df-haus 21490 df-tx 21736 df-hmeo 21929 df-fil 22020 df-fm 22112 df-flim 22113 df-flf 22114 df-xms 22495 df-ms 22496 df-tms 22497 df-cfil 23423 df-cau 23424 df-cmet 23425 df-grpo 27892 df-gid 27893 df-ginv 27894 df-gdiv 27895 df-ablo 27944 df-vc 27958 df-nv 27991 df-va 27994 df-ba 27995 df-sm 27996 df-0v 27997 df-vs 27998 df-nmcv 27999 df-ims 28000 df-dip 28100 df-ssp 28121 df-ph 28212 df-cbn 28263 df-hnorm 28369 df-hba 28370 df-hvsub 28372 df-hlim 28373 df-hcau 28374 df-sh 28608 df-ch 28622 df-oc 28653 df-ch0 28654 df-shs 28711 df-chj 28713 df-dmd 29684 |
This theorem is referenced by: dmdi4 29710 dmdbr5 29711 sumdmdi 29823 dmdbr4ati 29824 |
Copyright terms: Public domain | W3C validator |