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Mirrors > Home > MPE Home > Th. List > subidi | Structured version Visualization version GIF version |
Description: Subtraction of a number from itself. (Contributed by NM, 26-Nov-1994.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
Ref | Expression |
---|---|
subidi | ⊢ (𝐴 − 𝐴) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 | . 2 ⊢ 𝐴 ∈ ℂ | |
2 | subid 11515 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 − 𝐴) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7424 ℂcc 11142 0cc0 11144 − cmin 11480 |
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 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-po 5592 df-so 5593 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8729 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-ltxr 11289 df-sub 11482 |
This theorem is referenced by: 0reALT 11593 1m1e0 12320 0m0e0 12368 divalglem2 16377 divalglem9 16383 psgnunilem2 19455 psdmul 22095 pcoass 24969 sincosq1sgn 26451 resinf1o 26488 acos1 26845 bposlem2 27236 clwlkclwwlklem2a4 29825 2clwwlk2 30176 numclwlk1lem1 30197 bcseqi 30948 lnfn0i 31870 ballotth 34162 areaquad 42647 wallispilem4 45458 fouriersw 45621 2t6m3t4e0 47463 zlmodzxzequa 47615 zlmodzxzequap 47618 |
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