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Mirrors > Home > ILE Home > Th. List > npcan | GIF version |
Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
npcan | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcl 8186 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
2 | simpr 110 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
3 | 1, 2 | addcomd 8138 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = (𝐵 + (𝐴 − 𝐵))) |
4 | pncan3 8195 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 + (𝐴 − 𝐵)) = 𝐴) | |
5 | 4 | ancoms 268 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + (𝐴 − 𝐵)) = 𝐴) |
6 | 3, 5 | eqtrd 2222 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∈ wcel 2160 (class class class)co 5896 ℂcc 7839 + caddc 7844 − cmin 8158 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-resscn 7933 ax-1cn 7934 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-distr 7945 ax-i2m1 7946 ax-0id 7949 ax-rnegex 7950 ax-cnre 7952 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5899 df-oprab 5900 df-mpo 5901 df-sub 8160 |
This theorem is referenced by: addsubass 8197 npncan 8208 nppcan 8209 nnpcan 8210 subcan2 8212 nnncan 8222 npcand 8302 nn1suc 8968 zlem1lt 9339 zltlem1 9340 peano5uzti 9391 nummac 9458 uzp1 9591 peano2uzr 9615 fz01en 10083 fzsuc2 10109 fseq1m1p1 10125 fzoss2 10202 fzoaddel2 10223 fzosplitsnm1 10239 fzosplitprm1 10264 modfzo0difsn 10426 seq3m1 10499 monoord2 10508 ser3mono 10509 expm1t 10579 expubnd 10608 bcm1k 10772 bcn2 10776 hashfzo 10834 seq3coll 10854 shftlem 10857 shftfvalg 10859 shftfval 10862 iserex 11379 serf0 11392 fsumm1 11456 mptfzshft 11482 binomlem 11523 binom1dif 11527 isumsplit 11531 dvdssub2 11874 4sqlem19 12441 |
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