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Mirrors > Home > MPE Home > Th. List > div1 | Structured version Visualization version 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 | mullid 11259 | . 2 ⊢ (𝐴 ∈ ℂ → (1 · 𝐴) = 𝐴) | |
2 | ax-1cn 11212 | . . . . 5 ⊢ 1 ∈ ℂ | |
3 | ax-1ne0 11223 | . . . . 5 ⊢ 1 ≠ 0 | |
4 | 2, 3 | pm3.2i 469 | . . . 4 ⊢ (1 ∈ ℂ ∧ 1 ≠ 0) |
5 | divmul 11922 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (1 ∈ ℂ ∧ 1 ≠ 0)) → ((𝐴 / 1) = 𝐴 ↔ (1 · 𝐴) = 𝐴)) | |
6 | 4, 5 | mp3an3 1446 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((𝐴 / 1) = 𝐴 ↔ (1 · 𝐴) = 𝐴)) |
7 | 6 | anidms 565 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 / 1) = 𝐴 ↔ (1 · 𝐴) = 𝐴)) |
8 | 1, 7 | mpbird 256 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 / 1) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7423 ℂcc 11152 0cc0 11154 1c1 11155 · cmul 11159 / cdiv 11917 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 ax-resscn 11211 ax-1cn 11212 ax-icn 11213 ax-addcl 11214 ax-addrcl 11215 ax-mulcl 11216 ax-mulrcl 11217 ax-mulcom 11218 ax-addass 11219 ax-mulass 11220 ax-distr 11221 ax-i2m1 11222 ax-1ne0 11223 ax-1rid 11224 ax-rnegex 11225 ax-rrecex 11226 ax-cnre 11227 ax-pre-lttri 11228 ax-pre-lttrn 11229 ax-pre-ltadd 11230 ax-pre-mulgt0 11231 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-po 5593 df-so 5594 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7379 df-ov 7426 df-oprab 7427 df-mpo 7428 df-er 8733 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11296 df-mnf 11297 df-xr 11298 df-ltxr 11299 df-le 11300 df-sub 11492 df-neg 11493 df-div 11918 |
This theorem is referenced by: 1div1e1 11951 divdiv1 11972 divdiv2 11973 div1i 11989 div1d 12029 ef4p 16110 efgt1p2 16111 efgt1p 16112 dveflem 25994 logneg2 26634 |
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