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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 10975 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐵 − 𝐴) ∈ ℂ) | |
2 | eqid 2739 | . . . 4 ⊢ (𝐵 − 𝐴) = (𝐵 − 𝐴) | |
3 | subadd 10979 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (𝐵 − 𝐴) ∈ ℂ) → ((𝐵 − 𝐴) = (𝐵 − 𝐴) ↔ (𝐴 + (𝐵 − 𝐴)) = 𝐵)) | |
4 | 2, 3 | mpbii 236 | . . 3 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (𝐵 − 𝐴) ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
5 | 1, 4 | mpd3an3 1463 | . 2 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
6 | 5 | ancoms 462 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 (class class class)co 7182 ℂcc 10625 + caddc 10630 − cmin 10960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-po 5452 df-so 5453 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-er 8332 df-en 8568 df-dom 8569 df-sdom 8570 df-pnf 10767 df-mnf 10768 df-ltxr 10770 df-sub 10962 |
This theorem is referenced by: npcan 10985 nncan 11005 npncan3 11014 negid 11023 pncan3i 11053 pncan3d 11090 subdi 11163 posdif 11223 fzonmapblen 13186 fzen2 13440 bernneq2 13695 hashdom 13844 hashfz 13892 hashreshashfun 13904 swrdfv2 14124 addlenpfx 14154 ccatpfx 14164 2cshwid 14277 cshweqdif2 14282 2cshwcshw 14288 cshwcshid 14290 isercoll2 15130 isumshft 15299 dvdssubr 15762 vdwlem3 16431 vdwlem9 16437 prmgaplem7 16505 mplsubrglem 20832 blcvx 23562 dvef 24744 dvcvx 24784 sincosq2sgn 25256 sincosq3sgn 25257 sincosq4sgn 25258 eflogeq 25357 logdivlti 25375 advlogexp 25410 cvxcl 25734 scvxcvx 25735 cvxsconn 32788 resconn 32791 cos2h 35423 ftc1anclem5 35509 jm2.26a 40434 jm2.27c 40441 goldbachthlem1 44578 nn0sumshdiglemB 45547 |
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