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Mirrors > Home > MPE Home > Th. List > Mathboxes > zlmdsOLD | Structured version Visualization version GIF version |
Description: Obsolete proof of zlmds 32600 as of 11-Nov-2024. Distance in a ℤ -module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
zlmlem2.1 | ⊢ 𝑊 = (ℤMod‘𝐺) |
zlmds.1 | ⊢ 𝐷 = (dist‘𝐺) |
Ref | Expression |
---|---|
zlmdsOLD | ⊢ (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 2733 | . . . . 5 ⊢ (.g‘𝐺) = (.g‘𝐺) | |
4 | 2, 3 | zlmval 20932 | . . . 4 ⊢ (𝐺 ∈ 𝑉 → 𝑊 = ((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩)) |
5 | 4 | fveq2d 6847 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (dist‘𝑊) = (dist‘((𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩) sSet ⟨( ·𝑠 ‘ndx), (.g‘𝐺)⟩))) |
6 | dsid 17272 | . . . . 5 ⊢ dist = Slot (dist‘ndx) | |
7 | 5re 12245 | . . . . . . 7 ⊢ 5 ∈ ℝ | |
8 | 1nn 12169 | . . . . . . . 8 ⊢ 1 ∈ ℕ | |
9 | 2nn0 12435 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
10 | 5nn0 12438 | . . . . . . . 8 ⊢ 5 ∈ ℕ0 | |
11 | 5lt10 12758 | . . . . . . . 8 ⊢ 5 < ;10 | |
12 | 8, 9, 10, 11 | declti 12661 | . . . . . . 7 ⊢ 5 < ;12 |
13 | 7, 12 | gtneii 11272 | . . . . . 6 ⊢ ;12 ≠ 5 |
14 | dsndx 17271 | . . . . . . 7 ⊢ (dist‘ndx) = ;12 | |
15 | scandx 17200 | . . . . . . 7 ⊢ (Scalar‘ndx) = 5 | |
16 | 14, 15 | neeq12i 3007 | . . . . . 6 ⊢ ((dist‘ndx) ≠ (Scalar‘ndx) ↔ ;12 ≠ 5) |
17 | 13, 16 | mpbir 230 | . . . . 5 ⊢ (dist‘ndx) ≠ (Scalar‘ndx) |
18 | 6, 17 | setsnid 17086 | . . . 4 ⊢ (dist‘𝐺) = (dist‘(𝐺 sSet ⟨(Scalar‘ndx), ℤring⟩)) |
19 | 6re 12248 | . . . . . . 7 ⊢ 6 ∈ ℝ | |
20 | 6nn0 12439 | . . . . . . . 8 ⊢ 6 ∈ ℕ0 | |
21 | 6lt10 12757 | . . . . . . . 8 ⊢ 6 < ;10 | |
22 | 8, 9, 20, 21 | declti 12661 | . . . . . . 7 ⊢ 6 < ;12 |
23 | 19, 22 | gtneii 11272 | . . . . . 6 ⊢ ;12 ≠ 6 |
24 | vscandx 17205 | . . . . . . 7 ⊢ ( ·𝑠 ‘ndx) = 6 | |
25 | 14, 24 | neeq12i 3007 | . . . . . 6 ⊢ ((dist‘ndx) ≠ ( ·𝑠 ‘ndx) ↔ ;12 ≠ 6) |
26 | 23, 25 | mpbir 230 | . . . . 5 ⊢ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) |
27 | 6, 26 | setsnid 17086 | . . . 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 2107 ≠ wne 2940 ⟨cop 4593 ‘cfv 6497 (class class class)co 7358 1c1 11057 2c2 12213 5c5 12216 6c6 12217 ;cdc 12623 sSet csts 17040 ndxcnx 17070 Scalarcsca 17141 ·𝑠 cvsca 17142 distcds 17147 .gcmg 18877 ℤringczring 20885 ℤModczlm 20917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-sets 17041 df-slot 17059 df-ndx 17071 df-sca 17154 df-vsca 17155 df-ds 17160 df-zlm 20921 |
This theorem is referenced by: (None) |
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