pncan2d - Metamath Proof Explorer

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Theorem pncan2d 11604

Description: Cancellation law for subtraction. (Contributed by Mario Carneiro, 27-May-2016.)

**Hypotheses**
Ref	Expression
negidd.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
pncand.2	⊢ (𝜑 → 𝐵 ∈ ℂ)

**Assertion**
Ref	Expression
pncan2d	⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵)

**Proof of Theorem pncan2d**
Step	Hyp	Ref	Expression
1		negidd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		pncand.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		pncan2 11498	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐴) = 𝐵)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → ((𝐴 + 𝐵) − 𝐴) = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 (class class class)co 7420 ℂcc 11137 + caddc 11142 − cmin 11475

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-sub 11477

This theorem is referenced by: xov1plusxeqvd 13508 fzocatel 13729 expaddzlem 14103 hashf1lem2 14450 imval2 15131 clim2ser 15634 serf0 15660 fsumrev2 15761 geolim2 15850 mertenslem2 15864 bpolydiflem 16031 dvdsadd2b 16283 sadadd3 16436 mulgdirlem 19060 cnsubrg 21360 coe1tmmul2fv 22197 coe1pwmulfv 22199 reperflem 24747 reconnlem2 24756 ioorcl2 25514 uniioombllem3 25527 lhop1lem 25959 dvfsumabs 25970 ftc1lem1 25983 itgparts 25995 itgsubstlem 25996 coe1mul3 26048 coemulhi 26201 abelthlem6 26386 efif1olem4 26492 efopn 26605 dcubic2 26789 birthdaylem2 26897 lgamcvg2 27000 chtdif 27103 lgsquadlem1 27326 2sqmod 27382 dchrisumlem1 27435 dchrisumlem2 27436 dchrisum0lem1b 27461 pntrlog2bndlem1 27523 pntrlog2bndlem2 27524 axsegconlem9 28749 axpaschlem 28764 cycpmco2lem3 32862 cycpmco2lem4 32863 cycpmco2lem5 32864 cycpmco2lem6 32865 cycpmco2 32867 archiabllem1a 32912 probdif 34040 ballotlemsi 34134 knoppndvlem14 36000 knoppndvlem16 36002 bj-bary1lem1 36790 ftc1anc 37174 sticksstones12 41630 bcle2d 41651 readdrcl2d 41848 lsubrotld 41852 sumcubes 41873 jm2.27c 42428 jm3.1lem2 42439 radcnvrat 43751 binomcxplemdvbinom 43790 binomcxplemnotnn0 43793 mccllem 44985 ioodvbdlimc1lem2 45320 stirlinglem5 45466 fourierdlem7 45502 fourierdlem19 45514 fourierdlem26 45521 fourierdlem42 45537 fourierdlem63 45557 fourierdlem65 45559 fourierdlem79 45573 fourierdlem89 45583 fourierdlem90 45584 fourierdlem91 45585 fourierdlem101 45595 fourierdlem112 45606 qndenserrnbllem 45682

	
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