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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmds | Structured version Visualization version GIF version |
Description: Distance in a ℤ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
zlmlem2.1 | ⊢ 𝑊 = (ℤMod‘𝐺) |
zlmds.1 | ⊢ 𝐷 = (dist‘𝐺) |
Ref | Expression |
---|---|
zlmds | ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zlmlem2.1 | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
2 | eqid 2771 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | zlmval 20079 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
4 | 3 | fveq2d 6336 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (dist‘𝑊) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
5 | dsid 16271 | . . . . 5 ⊢ dist = Slot (dist‘ndx) | |
6 | 5re 11301 | . . . . . . 7 ⊢ 5 ∈ ℝ | |
7 | 1nn 11233 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
8 | 2nn0 11511 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
9 | 5nn0 11514 | . . . . . . . 8 ⊢ 5 ∈ ℕ0 | |
10 | 5lt10 11878 | . . . . . . . 8 ⊢ 5 < ;10 | |
11 | 7, 8, 9, 10 | declti 11748 | . . . . . . 7 ⊢ 5 < ;12 |
12 | 6, 11 | gtneii 10351 | . . . . . 6 ⊢ ;12 ≠ 5 |
13 | dsndx 16270 | . . . . . . 7 ⊢ (dist‘ndx) = ;12 | |
14 | scandx 16221 | . . . . . . 7 ⊢ (Scalar‘ndx) = 5 | |
15 | 13, 14 | neeq12i 3009 | . . . . . 6 ⊢ ((dist‘ndx) ≠ (Scalar‘ndx) ↔ ;12 ≠ 5) |
16 | 12, 15 | mpbir 221 | . . . . 5 ⊢ (dist‘ndx) ≠ (Scalar‘ndx) |
17 | 5, 16 | setsnid 16122 | . . . 4 ⊢ (dist‘𝐺) = (dist‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) |
18 | 6re 11303 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
19 | 6nn0 11515 | . . . . . . . 8 ⊢ 6 ∈ ℕ0 | |
20 | 6lt10 11877 | . . . . . . . 8 ⊢ 6 < ;10 | |
21 | 7, 8, 19, 20 | declti 11748 | . . . . . . 7 ⊢ 6 < ;12 |
22 | 18, 21 | gtneii 10351 | . . . . . 6 ⊢ ;12 ≠ 6 |
23 | vscandx 16223 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
24 | 13, 23 | neeq12i 3009 | . . . . . 6 ⊢ ((dist‘ndx) ≠ ( ·𝑠 ‘ndx) ↔ ;12 ≠ 6) |
25 | 22, 24 | mpbir 221 | . . . . 5 ⊢ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) |
26 | 5, 25 | setsnid 16122 | . . . 4 ⊢ (dist‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
27 | 17, 26 | eqtri 2793 | . . 3 ⊢ (dist‘𝐺) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
28 | 4, 27 | syl6eqr 2823 | . 2 ⊢ (𝐺 ∈ 𝑉 → (dist‘𝑊) = (dist‘𝐺)) |
29 | zlmds.1 | . 2 ⊢ 𝐷 = (dist‘𝐺) | |
30 | 28, 29 | syl6reqr 2824 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 〈cop 4322 ‘cfv 6031 (class class class)co 6793 1c1 10139 2c2 11272 5c5 11275 6c6 11276 ;cdc 11695 ndxcnx 16061 sSet csts 16062 Scalarcsca 16152 ·𝑠 cvsca 16153 distcds 16158 .gcmg 17748 ℤringzring 20033 ℤModczlm 20064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-7 11286 df-8 11287 df-9 11288 df-n0 11495 df-z 11580 df-dec 11696 df-ndx 16067 df-slot 16068 df-sets 16071 df-sca 16165 df-vsca 16166 df-ds 16172 df-zlm 20068 |
This theorem is referenced by: zlmnm 30350 zhmnrg 30351 |
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