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 2818 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | zlmval 20591 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
4 | 3 | fveq2d 6667 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (dist‘𝑊) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
5 | dsid 16664 | . . . . 5 ⊢ dist = Slot (dist‘ndx) | |
6 | 5re 11712 | . . . . . . 7 ⊢ 5 ∈ ℝ | |
7 | 1nn 11637 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
8 | 2nn0 11902 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
9 | 5nn0 11905 | . . . . . . . 8 ⊢ 5 ∈ ℕ0 | |
10 | 5lt10 12221 | . . . . . . . 8 ⊢ 5 < ;10 | |
11 | 7, 8, 9, 10 | declti 12124 | . . . . . . 7 ⊢ 5 < ;12 |
12 | 6, 11 | gtneii 10740 | . . . . . 6 ⊢ ;12 ≠ 5 |
13 | dsndx 16663 | . . . . . . 7 ⊢ (dist‘ndx) = ;12 | |
14 | scandx 16620 | . . . . . . 7 ⊢ (Scalar‘ndx) = 5 | |
15 | 13, 14 | neeq12i 3079 | . . . . . 6 ⊢ ((dist‘ndx) ≠ (Scalar‘ndx) ↔ ;12 ≠ 5) |
16 | 12, 15 | mpbir 232 | . . . . 5 ⊢ (dist‘ndx) ≠ (Scalar‘ndx) |
17 | 5, 16 | setsnid 16527 | . . . 4 ⊢ (dist‘𝐺) = (dist‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) |
18 | 6re 11715 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
19 | 6nn0 11906 | . . . . . . . 8 ⊢ 6 ∈ ℕ0 | |
20 | 6lt10 12220 | . . . . . . . 8 ⊢ 6 < ;10 | |
21 | 7, 8, 19, 20 | declti 12124 | . . . . . . 7 ⊢ 6 < ;12 |
22 | 18, 21 | gtneii 10740 | . . . . . 6 ⊢ ;12 ≠ 6 |
23 | vscandx 16622 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
24 | 13, 23 | neeq12i 3079 | . . . . . 6 ⊢ ((dist‘ndx) ≠ ( ·𝑠 ‘ndx) ↔ ;12 ≠ 6) |
25 | 22, 24 | mpbir 232 | . . . . 5 ⊢ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) |
26 | 5, 25 | setsnid 16527 | . . . 4 ⊢ (dist‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
27 | 17, 26 | eqtri 2841 | . . 3 ⊢ (dist‘𝐺) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
28 | 4, 27 | syl6eqr 2871 | . 2 ⊢ (𝐺 ∈ 𝑉 → (dist‘𝑊) = (dist‘𝐺)) |
29 | zlmds.1 | . 2 ⊢ 𝐷 = (dist‘𝐺) | |
30 | 28, 29 | syl6reqr 2872 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 〈cop 4563 ‘cfv 6348 (class class class)co 7145 1c1 10526 2c2 11680 5c5 11683 6c6 11684 ;cdc 12086 ndxcnx 16468 sSet csts 16469 Scalarcsca 16556 ·𝑠 cvsca 16557 distcds 16562 .gcmg 18162 ℤringzring 20545 ℤModczlm 20576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-ndx 16474 df-slot 16475 df-sets 16478 df-sca 16569 df-vsca 16570 df-ds 16575 df-zlm 20580 |
This theorem is referenced by: zlmnm 31106 zhmnrg 31107 |
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