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Mirrors > Home > MPE Home > Th. List > cjsub | Structured version Visualization version GIF version |
Description: Complex conjugate distributes over subtraction. (Contributed by NM, 28-Apr-2005.) |
Ref | Expression |
---|---|
cjsub | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 10880 | . . 3 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
2 | cjadd 14494 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ) → (∗‘(𝐴 + -𝐵)) = ((∗‘𝐴) + (∗‘-𝐵))) | |
3 | 1, 2 | sylan2 594 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 + -𝐵)) = ((∗‘𝐴) + (∗‘-𝐵))) |
4 | negsub 10928 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
5 | 4 | fveq2d 6669 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 + -𝐵)) = (∗‘(𝐴 − 𝐵))) |
6 | cjneg 14500 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (∗‘-𝐵) = -(∗‘𝐵)) | |
7 | 6 | adantl 484 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘-𝐵) = -(∗‘𝐵)) |
8 | 7 | oveq2d 7166 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) + (∗‘-𝐵)) = ((∗‘𝐴) + -(∗‘𝐵))) |
9 | cjcl 14458 | . . . 4 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
10 | cjcl 14458 | . . . 4 ⊢ (𝐵 ∈ ℂ → (∗‘𝐵) ∈ ℂ) | |
11 | negsub 10928 | . . . 4 ⊢ (((∗‘𝐴) ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → ((∗‘𝐴) + -(∗‘𝐵)) = ((∗‘𝐴) − (∗‘𝐵))) | |
12 | 9, 10, 11 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) + -(∗‘𝐵)) = ((∗‘𝐴) − (∗‘𝐵))) |
13 | 8, 12 | eqtrd 2856 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) + (∗‘-𝐵)) = ((∗‘𝐴) − (∗‘𝐵))) |
14 | 3, 5, 13 | 3eqtr3d 2864 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 + caddc 10534 − cmin 10864 -cneg 10865 ∗ccj 14449 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rmo 3146 df-rab 3147 df-v 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-po 5469 df-so 5470 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-2 11694 df-cj 14452 df-re 14453 df-im 14454 |
This theorem is referenced by: sqabssub 14637 cjcn2 14950 mul4sqlem 16283 dvcjbr 24540 isosctrlem2 25391 atancj 25482 dipsubdi 28620 his2sub2 28864 sigarmf 43104 |
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