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Mirrors > Home > ILE Home > Th. List > pncan | GIF version |
Description: Cancellation law for subtraction. (Contributed by NM, 10-May-2004.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
pncan | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 110 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ) | |
2 | simpl 109 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ) | |
3 | 1, 2 | addcomd 8106 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + 𝐴) = (𝐴 + 𝐵)) |
4 | addcl 7935 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | |
5 | subadd 8158 | . . 3 ⊢ (((𝐴 + 𝐵) ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (((𝐴 + 𝐵) − 𝐵) = 𝐴 ↔ (𝐵 + 𝐴) = (𝐴 + 𝐵))) | |
6 | 4, 1, 2, 5 | syl3anc 1238 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) − 𝐵) = 𝐴 ↔ (𝐵 + 𝐴) = (𝐴 + 𝐵))) |
7 | 3, 6 | mpbird 167 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 (class class class)co 5874 ℂcc 7808 + caddc 7813 − cmin 8126 |
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 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 ax-resscn 7902 ax-1cn 7903 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-addcom 7910 ax-addass 7912 ax-distr 7914 ax-i2m1 7915 ax-0id 7918 ax-rnegex 7919 ax-cnre 7921 |
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 4004 df-opab 4065 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-sub 8128 |
This theorem is referenced by: pncan2 8162 addsubass 8165 pncan3oi 8171 subid1 8175 nppcan2 8186 pncand 8267 nn1m1nn 8935 nnsub 8956 elnn0nn 9216 zrevaddcl 9301 nzadd 9303 elz2 9322 qrevaddcl 9642 irradd 9644 fzrev3 10084 elfzp1b 10094 fzrevral3 10104 fzval3 10201 subsq2 10624 bcp1nk 10737 bcp1m1 10740 bcpasc 10741 shftlem 10820 shftval5 10833 fsump1 11423 mptfzshft 11445 telfsumo 11469 fsumparts 11473 bcxmas 11492 isum1p 11495 geolim 11514 mertenslem2 11539 mertensabs 11540 eftlub 11693 effsumlt 11695 eirraplem 11779 dvdsadd 11838 prmind2 12114 fldivp1 12340 prmpwdvds 12347 pockthlem 12348 dvexp 14068 abssinper 14160 lgsvalmod 14313 2sqlem10 14354 |
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