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Mirrors > Home > ILE Home > Th. List > divcanap1d | GIF version |
Description: A cancellation law for division. (Contributed by Jim Kingdon, 29-Feb-2020.) |
Ref | Expression |
---|---|
divcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divclapd.3 | ⊢ (𝜑 → 𝐵 # 0) |
Ref | Expression |
---|---|
divcanap1d | ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | divclapd.3 | . 2 ⊢ (𝜑 → 𝐵 # 0) | |
4 | divcanap1 8209 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((𝐴 / 𝐵) · 𝐵) = 𝐴) | |
5 | 1, 2, 3, 4 | syl3anc 1175 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1290 ∈ wcel 1439 class class class wbr 3851 (class class class)co 5666 ℂcc 7409 0cc0 7411 · cmul 7416 # cap 8119 / cdiv 8200 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7497 ax-resscn 7498 ax-1cn 7499 ax-1re 7500 ax-icn 7501 ax-addcl 7502 ax-addrcl 7503 ax-mulcl 7504 ax-mulrcl 7505 ax-addcom 7506 ax-mulcom 7507 ax-addass 7508 ax-mulass 7509 ax-distr 7510 ax-i2m1 7511 ax-0lt1 7512 ax-1rid 7513 ax-0id 7514 ax-rnegex 7515 ax-precex 7516 ax-cnre 7517 ax-pre-ltirr 7518 ax-pre-ltwlin 7519 ax-pre-lttrn 7520 ax-pre-apti 7521 ax-pre-ltadd 7522 ax-pre-mulgt0 7523 ax-pre-mulext 7524 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rmo 2368 df-rab 2369 df-v 2622 df-sbc 2842 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-id 4129 df-po 4132 df-iso 4133 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-iota 4993 df-fun 5030 df-fv 5036 df-riota 5622 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-pnf 7585 df-mnf 7586 df-xr 7587 df-ltxr 7588 df-le 7589 df-sub 7716 df-neg 7717 df-reap 8113 df-ap 8120 df-div 8201 |
This theorem is referenced by: apdivmuld 8341 ltdiv23 8414 lediv23 8415 recp1lt1 8421 ledivp1 8425 xp1d2m1eqxm1d2 8729 div4p1lem1div2 8730 qmulz 9169 iccf1o 9482 bcpasc 10235 resqrexlemcalc1 10508 sqrtdiv 10536 geo2sum 10969 dvdsval2 11138 flodddiv4t2lthalf 11276 mulgcddvds 11415 qredeq 11417 isprm6 11465 sqrt2irrlem 11479 qmuldeneqnum 11512 hashgcdlem 11542 |
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