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Mirrors > Home > ILE Home > Th. List > subid1d | GIF version |
Description: Identity law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
subid1d | ⊢ (𝜑 → (𝐴 − 0) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | subid1 8128 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 − 0) = 𝐴) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 − 0) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 (class class class)co 5851 ℂcc 7761 0cc0 7763 − cmin 8079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-resscn 7855 ax-1cn 7856 ax-icn 7858 ax-addcl 7859 ax-addrcl 7860 ax-mulcl 7861 ax-addcom 7863 ax-addass 7865 ax-distr 7867 ax-i2m1 7868 ax-0id 7871 ax-rnegex 7872 ax-cnre 7874 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-sub 8081 |
This theorem is referenced by: suble0 8384 lesub0 8387 ltm1 8751 modqid 10294 modqeqmodmin 10339 bcn0 10678 bcnn 10680 hashfzo0 10747 hashfz0 10749 remul2 10826 max0addsup 11172 clim0c 11238 geolim 11463 addmodlteqALT 11808 dvdsmod 11811 ndvdssub 11878 nn0seqcvgd 11984 phiprmpw 12165 pczpre 12240 pcaddlem 12281 pcmpt2 12285 4sqlem9 12327 limcimolemlt 13388 dveflem 13442 sinmpi 13491 cosppi 13494 sinhalfpim 13497 sincosq2sgn 13503 apdifflemr 14041 |
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