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 | zlmds.1 | . 2 ⊢ 𝐷 = (dist‘𝐺) | |
2 | zlmlem2.1 | . . . . 5 ⊢ 𝑊 = (ℤMod‘𝐺) | |
3 | eqid 2738 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
4 | 2, 3 | zlmval 20329 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
5 | 4 | fveq2d 6672 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (dist‘𝑊) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
6 | dsid 16772 | . . . . 5 ⊢ dist = Slot (dist‘ndx) | |
7 | 5re 11796 | . . . . . . 7 ⊢ 5 ∈ ℝ | |
8 | 1nn 11720 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
9 | 2nn0 11986 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
10 | 5nn0 11989 | . . . . . . . 8 ⊢ 5 ∈ ℕ0 | |
11 | 5lt10 12307 | . . . . . . . 8 ⊢ 5 < ;10 | |
12 | 8, 9, 10, 11 | declti 12210 | . . . . . . 7 ⊢ 5 < ;12 |
13 | 7, 12 | gtneii 10823 | . . . . . 6 ⊢ ;12 ≠ 5 |
14 | dsndx 16771 | . . . . . . 7 ⊢ (dist‘ndx) = ;12 | |
15 | scandx 16728 | . . . . . . 7 ⊢ (Scalar‘ndx) = 5 | |
16 | 14, 15 | neeq12i 3000 | . . . . . 6 ⊢ ((dist‘ndx) ≠ (Scalar‘ndx) ↔ ;12 ≠ 5) |
17 | 13, 16 | mpbir 234 | . . . . 5 ⊢ (dist‘ndx) ≠ (Scalar‘ndx) |
18 | 6, 17 | setsnid 16635 | . . . 4 ⊢ (dist‘𝐺) = (dist‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) |
19 | 6re 11799 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
20 | 6nn0 11990 | . . . . . . . 8 ⊢ 6 ∈ ℕ0 | |
21 | 6lt10 12306 | . . . . . . . 8 ⊢ 6 < ;10 | |
22 | 8, 9, 20, 21 | declti 12210 | . . . . . . 7 ⊢ 6 < ;12 |
23 | 19, 22 | gtneii 10823 | . . . . . 6 ⊢ ;12 ≠ 6 |
24 | vscandx 16730 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
25 | 14, 24 | neeq12i 3000 | . . . . . 6 ⊢ ((dist‘ndx) ≠ ( ·𝑠 ‘ndx) ↔ ;12 ≠ 6) |
26 | 23, 25 | mpbir 234 | . . . . 5 ⊢ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) |
27 | 6, 26 | setsnid 16635 | . . . 4 ⊢ (dist‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
28 | 18, 27 | eqtri 2761 | . . 3 ⊢ (dist‘𝐺) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
29 | 5, 28 | eqtr4di 2791 | . 2 ⊢ (𝐺 ∈ 𝑉 → (dist‘𝑊) = (dist‘𝐺)) |
30 | 1, 29 | eqtr4id 2792 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 〈cop 4519 ‘cfv 6333 (class class class)co 7164 1c1 10609 2c2 11764 5c5 11767 6c6 11768 ;cdc 12172 ndxcnx 16576 sSet csts 16577 Scalarcsca 16664 ·𝑠 cvsca 16665 distcds 16670 .gcmg 18335 ℤringzring 20282 ℤModczlm 20314 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-ndx 16582 df-slot 16583 df-sets 16586 df-sca 16677 df-vsca 16678 df-ds 16683 df-zlm 20318 |
This theorem is referenced by: zlmnm 31478 zhmnrg 31479 |
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