Mathbox for Saveliy Skresanov |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sigarls | Structured version Visualization version GIF version |
Description: Signed area is linear by the second argument. (Contributed by Saveliy Skresanov, 19-Sep-2017.) |
Ref | Expression |
---|---|
sigar | ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) |
Ref | Expression |
---|---|
sigarls | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (𝐴𝐺(𝐵 · 𝐶)) = ((𝐴𝐺𝐵) · 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1132 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → 𝐴 ∈ ℂ) | |
2 | 1 | cjcld 14555 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (∗‘𝐴) ∈ ℂ) |
3 | simp2 1133 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℂ) | |
4 | simpr 487 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
5 | 4 | recnd 10669 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ) |
6 | 5 | 3adant1 1126 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℂ) |
7 | 2, 3, 6 | mulassd 10664 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (((∗‘𝐴) · 𝐵) · 𝐶) = ((∗‘𝐴) · (𝐵 · 𝐶))) |
8 | 7 | fveq2d 6674 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (ℑ‘(((∗‘𝐴) · 𝐵) · 𝐶)) = (ℑ‘((∗‘𝐴) · (𝐵 · 𝐶)))) |
9 | simp3 1134 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → 𝐶 ∈ ℝ) | |
10 | 2, 3 | mulcld 10661 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → ((∗‘𝐴) · 𝐵) ∈ ℂ) |
11 | 9, 10 | immul2d 14587 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (ℑ‘(𝐶 · ((∗‘𝐴) · 𝐵))) = (𝐶 · (ℑ‘((∗‘𝐴) · 𝐵)))) |
12 | 10, 6 | mulcomd 10662 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (((∗‘𝐴) · 𝐵) · 𝐶) = (𝐶 · ((∗‘𝐴) · 𝐵))) |
13 | 12 | fveq2d 6674 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (ℑ‘(((∗‘𝐴) · 𝐵) · 𝐶)) = (ℑ‘(𝐶 · ((∗‘𝐴) · 𝐵)))) |
14 | imcl 14470 | . . . . . . 7 ⊢ (((∗‘𝐴) · 𝐵) ∈ ℂ → (ℑ‘((∗‘𝐴) · 𝐵)) ∈ ℝ) | |
15 | 14 | recnd 10669 | . . . . . 6 ⊢ (((∗‘𝐴) · 𝐵) ∈ ℂ → (ℑ‘((∗‘𝐴) · 𝐵)) ∈ ℂ) |
16 | 10, 15 | syl 17 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (ℑ‘((∗‘𝐴) · 𝐵)) ∈ ℂ) |
17 | 16, 6 | mulcomd 10662 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → ((ℑ‘((∗‘𝐴) · 𝐵)) · 𝐶) = (𝐶 · (ℑ‘((∗‘𝐴) · 𝐵)))) |
18 | 11, 13, 17 | 3eqtr4d 2866 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (ℑ‘(((∗‘𝐴) · 𝐵) · 𝐶)) = ((ℑ‘((∗‘𝐴) · 𝐵)) · 𝐶)) |
19 | 8, 18 | eqtr3d 2858 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (ℑ‘((∗‘𝐴) · (𝐵 · 𝐶))) = ((ℑ‘((∗‘𝐴) · 𝐵)) · 𝐶)) |
20 | simpl 485 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → 𝐵 ∈ ℂ) | |
21 | 20, 5 | mulcld 10661 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) ∈ ℂ) |
22 | 21 | 3adant1 1126 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (𝐵 · 𝐶) ∈ ℂ) |
23 | sigar | . . . 4 ⊢ 𝐺 = (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (ℑ‘((∗‘𝑥) · 𝑦))) | |
24 | 23 | sigarval 43127 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 · 𝐶) ∈ ℂ) → (𝐴𝐺(𝐵 · 𝐶)) = (ℑ‘((∗‘𝐴) · (𝐵 · 𝐶)))) |
25 | 1, 22, 24 | syl2anc 586 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (𝐴𝐺(𝐵 · 𝐶)) = (ℑ‘((∗‘𝐴) · (𝐵 · 𝐶)))) |
26 | 23 | sigarval 43127 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵))) |
27 | 26 | 3adant3 1128 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (𝐴𝐺𝐵) = (ℑ‘((∗‘𝐴) · 𝐵))) |
28 | 27 | oveq1d 7171 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → ((𝐴𝐺𝐵) · 𝐶) = ((ℑ‘((∗‘𝐴) · 𝐵)) · 𝐶)) |
29 | 19, 25, 28 | 3eqtr4d 2866 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℝ) → (𝐴𝐺(𝐵 · 𝐶)) = ((𝐴𝐺𝐵) · 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 ℂcc 10535 ℝcr 10536 · cmul 10542 ∗ccj 14455 ℑcim 14457 |
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 ax-pre-mulgt0 10614 |
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-rmo 3146 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-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-2 11701 df-cj 14458 df-re 14459 df-im 14460 |
This theorem is referenced by: sigarcol 43141 sharhght 43142 sigaradd 43143 |
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