Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > srads | Structured version Visualization version GIF version |
Description: Distance function of a subring algebra. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 29-Oct-2024.) |
Ref | Expression |
---|---|
srapart.a | ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) |
srapart.s | ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) |
Ref | Expression |
---|---|
srads | ⊢ (𝜑 → (dist‘𝑊) = (dist‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srapart.a | . 2 ⊢ (𝜑 → 𝐴 = ((subringAlg ‘𝑊)‘𝑆)) | |
2 | srapart.s | . 2 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑊)) | |
3 | dsid 17094 | . 2 ⊢ dist = Slot (dist‘ndx) | |
4 | slotsdnscsi 17100 | . . . 4 ⊢ ((dist‘ndx) ≠ (Scalar‘ndx) ∧ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) ∧ (dist‘ndx) ≠ (·𝑖‘ndx)) | |
5 | 4 | simp1i 1138 | . . 3 ⊢ (dist‘ndx) ≠ (Scalar‘ndx) |
6 | 5 | necomi 2998 | . 2 ⊢ (Scalar‘ndx) ≠ (dist‘ndx) |
7 | 4 | simp2i 1139 | . . 3 ⊢ (dist‘ndx) ≠ ( ·𝑠 ‘ndx) |
8 | 7 | necomi 2998 | . 2 ⊢ ( ·𝑠 ‘ndx) ≠ (dist‘ndx) |
9 | 4 | simp3i 1140 | . . 3 ⊢ (dist‘ndx) ≠ (·𝑖‘ndx) |
10 | 9 | necomi 2998 | . 2 ⊢ (·𝑖‘ndx) ≠ (dist‘ndx) |
11 | 1, 2, 3, 6, 8, 10 | sralem 20437 | 1 ⊢ (𝜑 → (dist‘𝑊) = (dist‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ≠ wne 2943 ⊆ wss 3888 ‘cfv 6435 ndxcnx 16892 Basecbs 16910 Scalarcsca 16963 ·𝑠 cvsca 16964 ·𝑖cip 16965 distcds 16969 subringAlg csra 20428 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12436 df-sets 16863 df-slot 16881 df-ndx 16893 df-sca 16976 df-vsca 16977 df-ip 16978 df-ds 16982 df-sra 20432 |
This theorem is referenced by: rlmds 20472 sranlm 23846 srabn 24522 |
Copyright terms: Public domain | W3C validator |