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Mirrors > Home > ILE Home > Th. List > subidd | GIF version |
Description: Subtraction of a number from itself. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
Ref | Expression |
---|---|
subidd | ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | subid 8172 | . 2 ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐴 − 𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 (class class class)co 5872 ℂcc 7806 0cc0 7808 − cmin 8124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-setind 4535 ax-resscn 7900 ax-1cn 7901 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-addcom 7908 ax-addass 7910 ax-distr 7912 ax-i2m1 7913 ax-0id 7916 ax-rnegex 7917 ax-cnre 7919 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-iota 5177 df-fun 5217 df-fv 5223 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-sub 8126 |
This theorem is referenced by: mul02 8340 leaddle0 8430 cru 8555 iccf1o 10000 fzocatel 10194 zmod10 10335 hashfzo 10795 hashfzp1 10797 resqrexlemnm 11020 bdtri 11241 climconst 11291 telfsumo 11467 fsumparts 11471 cvgratnnlemmn 11526 cvgratnnlemseq 11527 nn0seqcvgd 12033 pcmpt2 12334 cncfmptc 13953 limcimolemlt 14004 dvcnp2cntop 14034 |
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