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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 10885 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 − 𝐴) ∈ ℂ) | |
2 | eqid 2821 | . . . 4 ⊢ (𝐵 − 𝐴) = (𝐵 − 𝐴) | |
3 | subadd 10889 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (𝐵 − 𝐴) ∈ ℂ) → ((𝐵 − 𝐴) = (𝐵 − 𝐴) ↔ (𝐴 + (𝐵 − 𝐴)) = 𝐵)) | |
4 | 2, 3 | mpbii 235 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (𝐵 − 𝐴) ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
5 | 1, 4 | mpd3an3 1458 | . 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
6 | 5 | ancoms 461 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 + caddc 10540 − cmin 10870 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 |
This theorem is referenced by: npcan 10895 nncan 10915 npncan3 10924 negid 10933 pncan3i 10963 pncan3d 11000 subdi 11073 posdif 11133 fzonmapblen 13084 fzen2 13338 bernneq2 13592 hashdom 13741 hashfz 13789 hashreshashfun 13801 swrdfv2 14023 addlenpfx 14053 ccatpfx 14063 2cshwid 14176 cshweqdif2 14181 2cshwcshw 14187 cshwcshid 14189 isercoll2 15025 isumshft 15194 dvdssubr 15655 vdwlem3 16319 vdwlem9 16325 prmgaplem7 16393 mplsubrglem 20219 blcvx 23406 dvef 24577 dvcvx 24617 sincosq2sgn 25085 sincosq3sgn 25086 sincosq4sgn 25087 eflogeq 25185 logdivlti 25203 advlogexp 25238 cvxcl 25562 scvxcvx 25563 cvxsconn 32490 resconn 32493 cos2h 34898 ftc1anclem5 34986 jm2.26a 39617 jm2.27c 39624 goldbachthlem1 43727 nn0sumshdiglemB 44700 |
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