![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 8110 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 + 𝐴) = (𝐴 + 𝐵)) |
4 | addcl 7938 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | |
5 | subadd 8162 | . . 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 5877 ℂcc 7811 + caddc 7816 − cmin 8130 |
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 4123 ax-pow 4176 ax-pr 4211 ax-setind 4538 ax-resscn 7905 ax-1cn 7906 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-distr 7917 ax-i2m1 7918 ax-0id 7921 ax-rnegex 7922 ax-cnre 7924 |
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 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-iota 5180 df-fun 5220 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-sub 8132 |
This theorem is referenced by: pncan2 8166 addsubass 8169 pncan3oi 8175 subid1 8179 nppcan2 8190 pncand 8271 nn1m1nn 8939 nnsub 8960 elnn0nn 9220 zrevaddcl 9305 nzadd 9307 elz2 9326 qrevaddcl 9646 irradd 9648 fzrev3 10089 elfzp1b 10099 fzrevral3 10109 fzval3 10206 subsq2 10630 bcp1nk 10744 bcp1m1 10747 bcpasc 10748 shftlem 10827 shftval5 10840 fsump1 11430 mptfzshft 11452 telfsumo 11476 fsumparts 11480 bcxmas 11499 isum1p 11502 geolim 11521 mertenslem2 11546 mertensabs 11547 eftlub 11700 effsumlt 11702 eirraplem 11786 dvdsadd 11845 prmind2 12122 fldivp1 12348 prmpwdvds 12355 pockthlem 12356 dvexp 14260 abssinper 14352 lgsvalmod 14505 2sqlem10 14557 |
Copyright terms: Public domain | W3C validator |