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Mirrors > Home > MPE Home > Th. List > subeq0 | Structured version Visualization version GIF version |
Description: If the difference between two numbers is zero, they are equal. (Contributed by NM, 16-Nov-1999.) |
Ref | Expression |
---|---|
subeq0 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subid 11510 | . . . 4 ⊢ (𝐵 ∈ ℂ → (𝐵 − 𝐵) = 0) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 − 𝐵) = 0) |
3 | 2 | eqeq2d 2739 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ (𝐴 − 𝐵) = 0)) |
4 | subcan2 11516 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ 𝐴 = 𝐵)) | |
5 | 4 | 3anidm23 1419 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ 𝐴 = 𝐵)) |
6 | 3, 5 | bitr3d 281 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 (class class class)co 7420 ℂcc 11137 0cc0 11139 − cmin 11475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-sub 11477 |
This theorem is referenced by: subeq0i 11571 subeq0d 11610 subne0d 11611 subeq0ad 11612 mulcan1g 11898 div2sub 12070 cju 12239 nn0sub 12553 addmodlteq 13944 geoserg 15845 geolim 15849 geolim2 15850 georeclim 15851 geoisum1c 15859 tanadd 16144 fzocongeq 16301 divalglem8 16377 mndodcongi 19498 odf1 19517 odf1o1 19527 cnmet 24701 iccpnfhmeo 24883 plyremlem 26252 geolim3 26287 abelthlem2 26382 abelthlem7 26388 efeq1 26475 tanregt0 26486 logtayl 26607 ang180lem1 26754 ang180lem2 26755 ang180lem3 26756 lawcos 26761 isosctrlem1 26763 isosctrlem2 26764 atandm2 26822 atandm4 26824 2efiatan 26863 tanatan 26864 dvatan 26880 mumullem2 27125 mersenne 27173 dchrsum2 27214 sumdchr2 27216 addsq2reu 27386 axcgrid 28740 axcontlem2 28789 hvmulcan2 30896 poimirlem13 37106 rencldnfilem 42240 qirropth 42328 dvconstbi 43771 isosctrlem1ALT 44373 rrx2pnedifcoorneor 47789 rrx2pnedifcoorneorr 47790 |
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