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 2610 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
3 | 1, 2 | zlmval 19683 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
4 | 3 | fveq2d 6107 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (dist‘𝑊) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉))) |
5 | dsid 15886 | . . . . 5 ⊢ dist = Slot (dist‘ndx) | |
6 | 5re 10976 | . . . . . . 7 ⊢ 5 ∈ ℝ | |
7 | 1nn 10908 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
8 | 2nn0 11186 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
9 | 5nn0 11189 | . . . . . . . 8 ⊢ 5 ∈ ℕ0 | |
10 | 5lt10 11553 | . . . . . . . 8 ⊢ 5 < ;10 | |
11 | 7, 8, 9, 10 | declti 11422 | . . . . . . 7 ⊢ 5 < ;12 |
12 | 6, 11 | gtneii 10028 | . . . . . 6 ⊢ ;12 ≠ 5 |
13 | dsndx 15885 | . . . . . . 7 ⊢ (dist‘ndx) = ;12 | |
14 | scandx 15836 | . . . . . . 7 ⊢ (Scalar‘ndx) = 5 | |
15 | 13, 14 | neeq12i 2848 | . . . . . 6 ⊢ ((dist‘ndx) ≠ (Scalar‘ndx) ↔ ;12 ≠ 5) |
16 | 12, 15 | mpbir 220 | . . . . 5 ⊢ (dist‘ndx) ≠ (Scalar‘ndx) |
17 | 5, 16 | setsnid 15743 | . . . 4 ⊢ (dist‘𝐺) = (dist‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) |
18 | 6re 10978 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
19 | 6nn0 11190 | . . . . . . . 8 ⊢ 6 ∈ ℕ0 | |
20 | 6lt10 11552 | . . . . . . . 8 ⊢ 6 < ;10 | |
21 | 7, 8, 19, 20 | declti 11422 | . . . . . . 7 ⊢ 6 < ;12 |
22 | 18, 21 | gtneii 10028 | . . . . . 6 ⊢ ;12 ≠ 6 |
23 | vscandx 15838 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
24 | 13, 23 | neeq12i 2848 | . . . . . 6 ⊢ ((dist‘ndx) ≠ ( ·𝑠 ‘ndx) ↔ ;12 ≠ 6) |
25 | 22, 24 | mpbir 220 | . . . . 5 ⊢ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) |
26 | 5, 25 | setsnid 15743 | . . . 4 ⊢ (dist‘(𝐺 sSet 〈(Scalar‘ndx), ℤring〉)) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
27 | 17, 26 | eqtri 2632 | . . 3 ⊢ (dist‘𝐺) = (dist‘((𝐺 sSet 〈(Scalar‘ndx), ℤring〉) sSet 〈( ·𝑠 ‘ndx), (.g‘𝐺)〉)) |
28 | 4, 27 | syl6eqr 2662 | . 2 ⊢ (𝐺 ∈ 𝑉 → (dist‘𝑊) = (dist‘𝐺)) |
29 | zlmds.1 | . 2 ⊢ 𝐷 = (dist‘𝐺) | |
30 | 28, 29 | syl6reqr 2663 | 1 ⊢ (𝐺 ∈ 𝑉 → 𝐷 = (dist‘𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 〈cop 4131 ‘cfv 5804 (class class class)co 6549 1c1 9816 2c2 10947 5c5 10950 6c6 10951 ;cdc 11369 ndxcnx 15692 sSet csts 15693 Scalarcsca 15771 ·𝑠 cvsca 15772 distcds 15777 .gcmg 17363 ℤringzring 19637 ℤModczlm 19668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-ndx 15698 df-slot 15699 df-sets 15701 df-sca 15784 df-vsca 15785 df-ds 15791 df-zlm 19672 |
This theorem is referenced by: zlmnm 29338 zhmnrg 29339 |
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