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| Mirrors > Home > MPE Home > Th. List > pncan3 | Structured version Visualization version GIF version | ||
| Description: Subtraction and addition of equals. (Contributed by NM, 14-Mar-2005.) (Proof shortened by Steven Nguyen, 8-Jan-2023.) |
| Ref | Expression |
|---|---|
| pncan3 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | subcl 11507 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 − 𝐴) ∈ ℂ) | |
| 2 | eqid 2737 | . . . 4 ⊢ (𝐵 − 𝐴) = (𝐵 − 𝐴) | |
| 3 | subadd 11511 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (𝐵 − 𝐴) ∈ ℂ) → ((𝐵 − 𝐴) = (𝐵 − 𝐴) ↔ (𝐴 + (𝐵 − 𝐴)) = 𝐵)) | |
| 4 | 2, 3 | mpbii 233 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (𝐵 − 𝐴) ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
| 5 | 1, 4 | mpd3an3 1464 | . 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
| 6 | 5 | ancoms 458 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 (class class class)co 7431 ℂcc 11153 + caddc 11158 − cmin 11492 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 |
| This theorem is referenced by: npcan 11517 nncan 11538 npncan3 11547 negid 11556 pncan3i 11586 pncan3d 11623 subdi 11696 posdif 11756 fzonmapblen 13748 fzen2 14010 bernneq2 14269 hashdom 14418 hashfz 14466 hashreshashfun 14478 swrdfv2 14699 addlenpfx 14729 ccatpfx 14739 2cshwid 14852 cshweqdif2 14857 2cshwcshw 14864 cshwcshid 14866 isercoll2 15705 isumshft 15875 dvdssubr 16342 vdwlem3 17021 vdwlem9 17027 prmgaplem7 17095 mplsubrglem 22024 blcvx 24819 dvef 26018 dvcvx 26059 sincosq2sgn 26541 sincosq3sgn 26542 sincosq4sgn 26543 eflogeq 26644 logdivlti 26662 advlogexp 26697 cvxcl 27028 scvxcvx 27029 cvxsconn 35248 resconn 35251 cos2h 37618 ftc1anclem5 37704 jm2.26a 43012 jm2.27c 43019 goldbachthlem1 47532 nn0sumshdiglemB 48541 |
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