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Mirrors > Home > MPE Home > Th. List > dividi | Structured version Visualization version GIF version |
Description: A number divided by itself is one. (Contributed by NM, 9-Feb-1995.) |
Ref | Expression |
---|---|
divclz.1 | ⊢ 𝐴 ∈ ℂ |
reccl.2 | ⊢ 𝐴 ≠ 0 |
Ref | Expression |
---|---|
dividi | ⊢ (𝐴 / 𝐴) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divclz.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | reccl.2 | . 2 ⊢ 𝐴 ≠ 0 | |
3 | divid 11130 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 / 𝐴) = 1) | |
4 | 1, 2, 3 | mp2an 679 | 1 ⊢ (𝐴 / 𝐴) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∈ wcel 2050 ≠ wne 2967 (class class class)co 6978 ℂcc 10335 0cc0 10337 1c1 10338 / cdiv 11100 |
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 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 |
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 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-op 4449 df-uni 4714 df-br 4931 df-opab 4993 df-mpt 5010 df-id 5313 df-po 5327 df-so 5328 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-er 8091 df-en 8309 df-dom 8310 df-sdom 8311 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 |
This theorem is referenced by: 2div2e1 11591 halfpm6th 11671 fldiv4p1lem1div2 13023 0.999... 15100 geoihalfsum 15101 efival 15368 ef01bndlem 15400 cos1bnd 15403 cos2bnd 15404 cos01gt0 15407 rpnnen2lem3 15432 rpnnen2lem11 15440 sincos4thpi 24805 tan4thpi 24806 sincos6thpi 24807 ang180lem1 25091 log2cnv 25227 log2tlbnd 25228 log2le1 25233 ppiub 25485 bposlem8 25572 2lgslem3c 25679 2lgslem3d 25680 2lgsoddprmlem3b 25692 dp2ltsuc 30311 ballotth 31441 quad3 32433 taupilem1 34044 areaquad 39219 lhe4.4ex1a 40077 stoweidlem26 41743 stoweidlem34 41751 stirlinglem3 41793 dirkercncflem1 41820 fourierdlem24 41848 fourierdlem95 41918 fourierdlem103 41926 fourierdlem104 41927 |
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