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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 10894 | . . . 4 ⊢ (𝐵 ∈ ℂ → (𝐵 − 𝐵) = 0) | |
2 | 1 | adantl 485 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 − 𝐵) = 0) |
3 | 2 | eqeq2d 2809 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ (𝐴 − 𝐵) = 0)) |
4 | subcan2 10900 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ 𝐴 = 𝐵)) | |
5 | 4 | 3anidm23 1418 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ 𝐴 = 𝐵)) |
6 | 3, 5 | bitr3d 284 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = 0 ↔ 𝐴 = 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 0cc0 10526 − cmin 10859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 |
This theorem is referenced by: subeq0i 10955 subeq0d 10994 subne0d 10995 subeq0ad 10996 mulcan1g 11282 div2sub 11454 cju 11621 nn0sub 11935 addmodlteq 13309 geoserg 15213 geolim 15218 geolim2 15219 georeclim 15220 geoisum1c 15228 tanadd 15512 fzocongeq 15666 divalglem8 15741 mndodcongi 18663 odf1 18681 odf1o1 18689 cnmet 23377 iccpnfhmeo 23550 plyremlem 24900 geolim3 24935 abelthlem2 25027 abelthlem7 25033 efeq1 25120 tanregt0 25131 logtayl 25251 ang180lem1 25395 ang180lem2 25396 ang180lem3 25397 lawcos 25402 isosctrlem1 25404 isosctrlem2 25405 atandm2 25463 atandm4 25465 2efiatan 25504 tanatan 25505 dvatan 25521 mumullem2 25765 mersenne 25811 dchrsum2 25852 sumdchr2 25854 addsq2reu 26024 axcgrid 26710 axcontlem2 26759 hvmulcan2 28856 poimirlem13 35070 rencldnfilem 39761 qirropth 39849 dvconstbi 41038 isosctrlem1ALT 41640 rrx2pnedifcoorneor 45130 rrx2pnedifcoorneorr 45131 |
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