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Mirrors > Home > ILE Home > Th. List > div1 | GIF version |
Description: A number divided by 1 is itself. (Contributed by NM, 9-Jan-2002.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
div1 | ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mulid2 7764 | . 2 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
2 | ax-1cn 7713 | . . . . 5 ⊢ 1 ∈ ℂ | |
3 | 1ap0 8352 | . . . . 5 ⊢ 1 # 0 | |
4 | 2, 3 | pm3.2i 270 | . . . 4 ⊢ (1 ∈ ℂ ∧ 1 # 0) |
5 | divmulap 8435 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (1 ∈ ℂ ∧ 1 # 0)) → ((𝐴 / 1) = 𝐴 ↔ (1 · 𝐴) = 𝐴)) | |
6 | 4, 5 | mp3an3 1304 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 / 1) = 𝐴 ↔ (1 · 𝐴) = 𝐴)) |
7 | 6 | anidms 394 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 1) = 𝐴 ↔ (1 · 𝐴) = 𝐴)) |
8 | 1, 7 | mpbird 166 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℂcc 7618 0cc0 7620 1c1 7621 · cmul 7625 # cap 8343 / cdiv 8432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 |
This theorem is referenced by: 1div1e1 8464 divdivap1 8483 divdivap2 8484 div1i 8500 div1d 8540 ef4p 11400 efgt1p2 11401 efgt1p 11402 dveflem 12855 |
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