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Mirrors > Home > ILE Home > Th. List > subid | GIF version |
Description: Subtraction of a number from itself. (Contributed by NM, 8-Oct-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
subid | ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addid1 8085 | . . 3 ⊢ (𝐴 ∈ ℂ → (𝐴 + 0) = 𝐴) | |
2 | 1 | oveq1d 5884 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 0) − 𝐴) = (𝐴 − 𝐴)) |
3 | 0cn 7940 | . . 3 ⊢ 0 ∈ ℂ | |
4 | pncan2 8154 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 0 ∈ ℂ) → ((𝐴 + 0) − 𝐴) = 0) | |
5 | 3, 4 | mpan2 425 | . 2 ⊢ (𝐴 ∈ ℂ → ((𝐴 + 0) − 𝐴) = 0) |
6 | 2, 5 | eqtr3d 2212 | 1 ⊢ (𝐴 ∈ ℂ → (𝐴 − 𝐴) = 0) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 ∈ wcel 2148 (class class class)co 5869 ℂcc 7800 0cc0 7802 + caddc 7805 − cmin 8118 |
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 4118 ax-pow 4171 ax-pr 4206 ax-setind 4533 ax-resscn 7894 ax-1cn 7895 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-sub 8120 |
This theorem is referenced by: subeq0 8173 npncan2 8174 neg0 8193 subidi 8218 subidd 8246 addid0 8320 fisum0diag2 11439 mulc1cncf 13743 dvconst 13828 dvef 13855 |
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