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Mirrors > Home > ILE Home > Th. List > negsubd | GIF version |
Description: Relationship between subtraction and negative. Theorem I.3 of [Apostol] p. 18. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
negidd.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
pncand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
Ref | Expression |
---|---|
negsubd | ⊢ (𝜑 → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | pncand.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | negsub 8137 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1342 ∈ wcel 2135 (class class class)co 5836 ℂcc 7742 + caddc 7747 − cmin 8060 -cneg 8061 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-setind 4508 ax-resscn 7836 ax-1cn 7837 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-addcom 7844 ax-addass 7846 ax-distr 7848 ax-i2m1 7849 ax-0id 7852 ax-rnegex 7853 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-sub 8062 df-neg 8063 |
This theorem is referenced by: mulsub 8290 apsub1 8531 divsubdirap 8595 divsubdivap 8615 div2subap 8724 zaddcllemneg 9221 icoshftf1o 9918 fzosubel 10119 ceiqm1l 10236 modqcyc2 10285 qnegmod 10294 modqsub12d 10306 modsumfzodifsn 10321 expaddzaplem 10488 binom2sub 10557 seq3shft 10766 cjreb 10794 recj 10795 remullem 10799 imcj 10803 resqrexlemover 10938 resqrexlemcalc1 10942 resqrexlemcalc3 10944 bdtri 11167 subcn2 11238 fsumshftm 11372 fsumsub 11379 geosergap 11433 efmival 11660 cosadd 11664 sinsub 11667 sincossq 11675 cos12dec 11694 moddvds 11725 dvdsadd2b 11765 pythagtriplem4 12179 dvmptsubcn 13232 cosq34lt1 13318 rpcxpsub 13376 rpabscxpbnd 13406 rprelogbdiv 13422 apdifflemr 13767 |
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