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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.) |
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 | df-ds 16443 | . 2 ⊢ dist = Slot ;12 | |
4 | 1nn0 11725 | . . 3 ⊢ 1 ∈ ℕ0 | |
5 | 2nn 11513 | . . 3 ⊢ 2 ∈ ℕ | |
6 | 4, 5 | decnncl 11932 | . 2 ⊢ ;12 ∈ ℕ |
7 | 1nn 11452 | . . . 4 ⊢ 1 ∈ ℕ | |
8 | 2nn0 11726 | . . . 4 ⊢ 2 ∈ ℕ0 | |
9 | 8nn0 11732 | . . . 4 ⊢ 8 ∈ ℕ0 | |
10 | 8lt10 12045 | . . . 4 ⊢ 8 < ;10 | |
11 | 7, 8, 9, 10 | declti 11950 | . . 3 ⊢ 8 < ;12 |
12 | 11 | olci 852 | . 2 ⊢ (;12 < 5 ∨ 8 < ;12) |
13 | 1, 2, 3, 6, 12 | sralem 19671 | 1 ⊢ (𝜑 → (dist‘𝑊) = (dist‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ⊆ wss 3830 class class class wbr 4929 ‘cfv 6188 1c1 10336 < clt 10474 2c2 11495 5c5 11498 8c8 11501 ;cdc 11911 Basecbs 16339 distcds 16430 subringAlg csra 19662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-4 11505 df-5 11506 df-6 11507 df-7 11508 df-8 11509 df-9 11510 df-n0 11708 df-z 11794 df-dec 11912 df-ndx 16342 df-slot 16343 df-sets 16346 df-sca 16437 df-vsca 16438 df-ip 16439 df-ds 16443 df-sra 19666 |
This theorem is referenced by: rlmds 19698 sranlm 22996 srabn 23666 |
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