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Mirrors > Home > HSE Home > Th. List > hatomistici | Structured version Visualization version GIF version |
Description: Cℋ is atomistic, i.e. any element is the supremum of its atoms. Remark in [Kalmbach] p. 140. (Contributed by NM, 14-Aug-2002.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hatomistic.1 | ⊢ 𝐴 ∈ Cℋ |
Ref | Expression |
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hatomistici | ⊢ 𝐴 = ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4073 | . . . . 5 ⊢ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ HAtoms | |
2 | atssch 31459 | . . . . 5 ⊢ HAtoms ⊆ Cℋ | |
3 | 1, 2 | sstri 3987 | . . . 4 ⊢ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ Cℋ |
4 | chsupcl 30456 | . . . 4 ⊢ ({𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ Cℋ → ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∈ Cℋ ) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∈ Cℋ |
6 | hatomistic.1 | . . . 4 ⊢ 𝐴 ∈ Cℋ | |
7 | 6 | chshii 30343 | . . 3 ⊢ 𝐴 ∈ Sℋ |
8 | atelch 31460 | . . . . . . . 8 ⊢ (𝑦 ∈ HAtoms → 𝑦 ∈ Cℋ ) | |
9 | 8 | anim1i 615 | . . . . . . 7 ⊢ ((𝑦 ∈ HAtoms ∧ 𝑦 ⊆ 𝐴) → (𝑦 ∈ Cℋ ∧ 𝑦 ⊆ 𝐴)) |
10 | sseq1 4003 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ 𝐴 ↔ 𝑦 ⊆ 𝐴)) | |
11 | 10 | elrab 3679 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ↔ (𝑦 ∈ HAtoms ∧ 𝑦 ⊆ 𝐴)) |
12 | 10 | elrab 3679 | . . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴} ↔ (𝑦 ∈ Cℋ ∧ 𝑦 ⊆ 𝐴)) |
13 | 9, 11, 12 | 3imtr4i 291 | . . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} → 𝑦 ∈ {𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴}) |
14 | 13 | ssriv 3982 | . . . . 5 ⊢ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ {𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴} |
15 | ssrab2 4073 | . . . . . 6 ⊢ {𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴} ⊆ Cℋ | |
16 | chsupss 30458 | . . . . . 6 ⊢ (({𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ Cℋ ∧ {𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴} ⊆ Cℋ ) → ({𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ {𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴} → ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ⊆ ( ∨ℋ ‘{𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴}))) | |
17 | 3, 15, 16 | mp2an 690 | . . . . 5 ⊢ ({𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ {𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴} → ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ⊆ ( ∨ℋ ‘{𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴})) |
18 | 14, 17 | ax-mp 5 | . . . 4 ⊢ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ⊆ ( ∨ℋ ‘{𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴}) |
19 | chsupid 30528 | . . . . 5 ⊢ (𝐴 ∈ Cℋ → ( ∨ℋ ‘{𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴}) = 𝐴) | |
20 | 6, 19 | ax-mp 5 | . . . 4 ⊢ ( ∨ℋ ‘{𝑥 ∈ Cℋ ∣ 𝑥 ⊆ 𝐴}) = 𝐴 |
21 | 18, 20 | sseqtri 4014 | . . 3 ⊢ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ⊆ 𝐴 |
22 | elssuni 4934 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} → 𝑦 ⊆ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) | |
23 | 11, 22 | sylbir 234 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ HAtoms ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) |
24 | chsupunss 30460 | . . . . . . . . . . 11 ⊢ ({𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ Cℋ → ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})) | |
25 | 3, 24 | ax-mp 5 | . . . . . . . . . 10 ⊢ ∪ {𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴} ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) |
26 | 23, 25 | sstrdi 3990 | . . . . . . . . 9 ⊢ ((𝑦 ∈ HAtoms ∧ 𝑦 ⊆ 𝐴) → 𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})) |
27 | 26 | ex 413 | . . . . . . . 8 ⊢ (𝑦 ∈ HAtoms → (𝑦 ⊆ 𝐴 → 𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) |
28 | atne0 31461 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ HAtoms → 𝑦 ≠ 0ℋ) | |
29 | 28 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ HAtoms ∧ 𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})) → 𝑦 ≠ 0ℋ) |
30 | ssin 4226 | . . . . . . . . . . . . . . 15 ⊢ ((𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∧ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) ↔ 𝑦 ⊆ (( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) | |
31 | 5 | chocini 30570 | . . . . . . . . . . . . . . . 16 ⊢ (( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) = 0ℋ |
32 | 31 | sseq2i 4007 | . . . . . . . . . . . . . . 15 ⊢ (𝑦 ⊆ (( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) ↔ 𝑦 ⊆ 0ℋ) |
33 | 30, 32 | bitr2i 275 | . . . . . . . . . . . . . 14 ⊢ (𝑦 ⊆ 0ℋ ↔ (𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∧ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) |
34 | chle0 30559 | . . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ Cℋ → (𝑦 ⊆ 0ℋ ↔ 𝑦 = 0ℋ)) | |
35 | 8, 34 | syl 17 | . . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ HAtoms → (𝑦 ⊆ 0ℋ ↔ 𝑦 = 0ℋ)) |
36 | 33, 35 | bitr3id 284 | . . . . . . . . . . . . 13 ⊢ (𝑦 ∈ HAtoms → ((𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∧ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) ↔ 𝑦 = 0ℋ)) |
37 | 36 | biimpa 477 | . . . . . . . . . . . 12 ⊢ ((𝑦 ∈ HAtoms ∧ (𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) ∧ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) → 𝑦 = 0ℋ) |
38 | 37 | expr 457 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ HAtoms ∧ 𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})) → (𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})) → 𝑦 = 0ℋ)) |
39 | 38 | necon3ad 2952 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ HAtoms ∧ 𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})) → (𝑦 ≠ 0ℋ → ¬ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) |
40 | 29, 39 | mpd 15 | . . . . . . . . 9 ⊢ ((𝑦 ∈ HAtoms ∧ 𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})) → ¬ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) |
41 | 40 | ex 413 | . . . . . . . 8 ⊢ (𝑦 ∈ HAtoms → (𝑦 ⊆ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) → ¬ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) |
42 | 27, 41 | syld 47 | . . . . . . 7 ⊢ (𝑦 ∈ HAtoms → (𝑦 ⊆ 𝐴 → ¬ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) |
43 | imnan 400 | . . . . . . 7 ⊢ ((𝑦 ⊆ 𝐴 → ¬ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) ↔ ¬ (𝑦 ⊆ 𝐴 ∧ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) | |
44 | 42, 43 | sylib 217 | . . . . . 6 ⊢ (𝑦 ∈ HAtoms → ¬ (𝑦 ⊆ 𝐴 ∧ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) |
45 | ssin 4226 | . . . . . 6 ⊢ ((𝑦 ⊆ 𝐴 ∧ 𝑦 ⊆ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) ↔ 𝑦 ⊆ (𝐴 ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) | |
46 | 44, 45 | sylnib 327 | . . . . 5 ⊢ (𝑦 ∈ HAtoms → ¬ 𝑦 ⊆ (𝐴 ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) |
47 | 46 | nrex 3073 | . . . 4 ⊢ ¬ ∃𝑦 ∈ HAtoms 𝑦 ⊆ (𝐴 ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) |
48 | 5 | choccli 30423 | . . . . . . 7 ⊢ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})) ∈ Cℋ |
49 | 6, 48 | chincli 30576 | . . . . . 6 ⊢ (𝐴 ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) ∈ Cℋ |
50 | 49 | hatomici 31475 | . . . . 5 ⊢ ((𝐴 ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) ≠ 0ℋ → ∃𝑦 ∈ HAtoms 𝑦 ⊆ (𝐴 ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴})))) |
51 | 50 | necon1bi 2968 | . . . 4 ⊢ (¬ ∃𝑦 ∈ HAtoms 𝑦 ⊆ (𝐴 ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) → (𝐴 ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) = 0ℋ) |
52 | 47, 51 | ax-mp 5 | . . 3 ⊢ (𝐴 ∩ (⊥‘( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}))) = 0ℋ |
53 | 5, 7, 21, 52 | omlsii 30519 | . 2 ⊢ ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) = 𝐴 |
54 | 53 | eqcomi 2740 | 1 ⊢ 𝐴 = ( ∨ℋ ‘{𝑥 ∈ HAtoms ∣ 𝑥 ⊆ 𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∃wrex 3069 {crab 3431 ∩ cin 3943 ⊆ wss 3944 ∪ cuni 4901 ‘cfv 6532 Cℋ cch 30045 ⊥cort 30046 ∨ℋ chsup 30050 0ℋc0h 30051 HAtomscat 30081 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cc 10412 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 ax-hilex 30115 ax-hfvadd 30116 ax-hvcom 30117 ax-hvass 30118 ax-hv0cl 30119 ax-hvaddid 30120 ax-hfvmul 30121 ax-hvmulid 30122 ax-hvmulass 30123 ax-hvdistr1 30124 ax-hvdistr2 30125 ax-hvmul0 30126 ax-hfi 30195 ax-his1 30198 ax-his2 30199 ax-his3 30200 ax-his4 30201 ax-hcompl 30318 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-oadd 8452 df-omul 8453 df-er 8686 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9487 df-card 9916 df-acn 9919 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-ioo 13310 df-ico 13312 df-icc 13313 df-fz 13467 df-fzo 13610 df-fl 13739 df-seq 13949 df-exp 14010 df-hash 14273 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-clim 15414 df-rlim 15415 df-sum 15615 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17350 df-topn 17351 df-0g 17369 df-gsum 17370 df-topgen 17371 df-pt 17372 df-prds 17375 df-xrs 17430 df-qtop 17435 df-imas 17436 df-xps 17438 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-mulg 18923 df-cntz 19147 df-cmn 19614 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-fbas 20875 df-fg 20876 df-cnfld 20879 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-cld 22452 df-ntr 22453 df-cls 22454 df-nei 22531 df-cn 22660 df-cnp 22661 df-lm 22662 df-haus 22748 df-tx 22995 df-hmeo 23188 df-fil 23279 df-fm 23371 df-flim 23372 df-flf 23373 df-xms 23755 df-ms 23756 df-tms 23757 df-cfil 24701 df-cau 24702 df-cmet 24703 df-grpo 29609 df-gid 29610 df-ginv 29611 df-gdiv 29612 df-ablo 29661 df-vc 29675 df-nv 29708 df-va 29711 df-ba 29712 df-sm 29713 df-0v 29714 df-vs 29715 df-nmcv 29716 df-ims 29717 df-dip 29817 df-ssp 29838 df-ph 29929 df-cbn 29979 df-hnorm 30084 df-hba 30085 df-hvsub 30087 df-hlim 30088 df-hcau 30089 df-sh 30323 df-ch 30337 df-oc 30368 df-ch0 30369 df-span 30425 df-chsup 30427 df-cv 31395 df-at 31454 |
This theorem is referenced by: chpssati 31479 |
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