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Mirrors > Home > HSE Home > Th. List > lnopunilem2 | Structured version Visualization version GIF version |
Description: Lemma for lnopunii 30783. (Contributed by NM, 12-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopunilem.1 | ⊢ 𝑇 ∈ LinOp |
lnopunilem.2 | ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) |
lnopunilem.3 | ⊢ 𝐴 ∈ ℋ |
lnopunilem.4 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
lnopunilem2 | ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvoveq1 7374 | . . . . 5 ⊢ (𝑦 = if(𝑦 ∈ ℂ, 𝑦, 0) → (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(if(𝑦 ∈ ℂ, 𝑦, 0) · ((𝑇‘𝐴) ·ih (𝑇‘𝐵))))) | |
2 | fvoveq1 7374 | . . . . 5 ⊢ (𝑦 = if(𝑦 ∈ ℂ, 𝑦, 0) → (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) = (ℜ‘(if(𝑦 ∈ ℂ, 𝑦, 0) · (𝐴 ·ih 𝐵)))) | |
3 | 1, 2 | eqeq12d 2753 | . . . 4 ⊢ (𝑦 = if(𝑦 ∈ ℂ, 𝑦, 0) → ((ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) ↔ (ℜ‘(if(𝑦 ∈ ℂ, 𝑦, 0) · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(if(𝑦 ∈ ℂ, 𝑦, 0) · (𝐴 ·ih 𝐵))))) |
4 | lnopunilem.1 | . . . . 5 ⊢ 𝑇 ∈ LinOp | |
5 | lnopunilem.2 | . . . . 5 ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) | |
6 | lnopunilem.3 | . . . . 5 ⊢ 𝐴 ∈ ℋ | |
7 | lnopunilem.4 | . . . . 5 ⊢ 𝐵 ∈ ℋ | |
8 | 0cn 11105 | . . . . . 6 ⊢ 0 ∈ ℂ | |
9 | 8 | elimel 4553 | . . . . 5 ⊢ if(𝑦 ∈ ℂ, 𝑦, 0) ∈ ℂ |
10 | 4, 5, 6, 7, 9 | lnopunilem1 30781 | . . . 4 ⊢ (ℜ‘(if(𝑦 ∈ ℂ, 𝑦, 0) · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(if(𝑦 ∈ ℂ, 𝑦, 0) · (𝐴 ·ih 𝐵))) |
11 | 3, 10 | dedth 4542 | . . 3 ⊢ (𝑦 ∈ ℂ → (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵)))) |
12 | 11 | rgen 3064 | . 2 ⊢ ∀𝑦 ∈ ℂ (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) |
13 | 4 | lnopfi 30740 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ |
14 | 13 | ffvelcdmi 7030 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
15 | 6, 14 | ax-mp 5 | . . . 4 ⊢ (𝑇‘𝐴) ∈ ℋ |
16 | 13 | ffvelcdmi 7030 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℋ) |
17 | 7, 16 | ax-mp 5 | . . . 4 ⊢ (𝑇‘𝐵) ∈ ℋ |
18 | 15, 17 | hicli 29852 | . . 3 ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) ∈ ℂ |
19 | 6, 7 | hicli 29852 | . . 3 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
20 | recan 15181 | . . 3 ⊢ ((((𝑇‘𝐴) ·ih (𝑇‘𝐵)) ∈ ℂ ∧ (𝐴 ·ih 𝐵) ∈ ℂ) → (∀𝑦 ∈ ℂ (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) ↔ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵))) | |
21 | 18, 19, 20 | mp2an 690 | . 2 ⊢ (∀𝑦 ∈ ℂ (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) ↔ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵)) |
22 | 12, 21 | mpbi 229 | 1 ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ifcif 4484 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 0cc0 11009 · cmul 11014 ℜcre 14942 ℋchba 29690 ·ih csp 29693 normℎcno 29694 LinOpclo 29718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-hilex 29770 ax-hfvadd 29771 ax-hv0cl 29774 ax-hfvmul 29776 ax-hvmul0 29781 ax-hfi 29850 ax-his1 29853 ax-his2 29854 ax-his3 29855 ax-his4 29856 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-sup 9336 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-n0 12372 df-z 12458 df-uz 12722 df-rp 12870 df-seq 13861 df-exp 13922 df-cj 14944 df-re 14945 df-im 14946 df-sqrt 15080 df-hnorm 29739 df-lnop 30612 |
This theorem is referenced by: lnopunii 30783 |
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