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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 11421 | . . . 4 ⊢ (𝐵 ∈ ℂ → (𝐵 − 𝐵) = 0) | |
2 | 1 | adantl 483 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 − 𝐵) = 0) |
3 | 2 | eqeq2d 2748 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ (𝐴 − 𝐵) = 0)) |
4 | subcan2 11427 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) = (𝐵 − 𝐵) ↔ 𝐴 = 𝐵)) | |
5 | 4 | 3anidm23 1422 | . 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 397 = wceq 1542 ∈ wcel 2107 (class class class)co 7358 ℂcc 11050 0cc0 11052 − cmin 11386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-ltxr 11195 df-sub 11388 |
This theorem is referenced by: subeq0i 11482 subeq0d 11521 subne0d 11522 subeq0ad 11523 mulcan1g 11809 div2sub 11981 cju 12150 nn0sub 12464 addmodlteq 13852 geoserg 15752 geolim 15756 geolim2 15757 georeclim 15758 geoisum1c 15766 tanadd 16050 fzocongeq 16207 divalglem8 16283 mndodcongi 19326 odf1 19345 odf1o1 19355 cnmet 24138 iccpnfhmeo 24311 plyremlem 25667 geolim3 25702 abelthlem2 25794 abelthlem7 25800 efeq1 25887 tanregt0 25898 logtayl 26018 ang180lem1 26162 ang180lem2 26163 ang180lem3 26164 lawcos 26169 isosctrlem1 26171 isosctrlem2 26172 atandm2 26230 atandm4 26232 2efiatan 26271 tanatan 26272 dvatan 26288 mumullem2 26532 mersenne 26578 dchrsum2 26619 sumdchr2 26621 addsq2reu 26791 axcgrid 27868 axcontlem2 27917 hvmulcan2 30018 poimirlem13 36094 rencldnfilem 41146 qirropth 41234 dvconstbi 42621 isosctrlem1ALT 43223 rrx2pnedifcoorneor 46809 rrx2pnedifcoorneorr 46810 |
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