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Mirrors > Home > HSE Home > Th. List > lnopunilem2 | Structured version Visualization version GIF version |
Description: Lemma for lnopunii 30093. (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 7236 | . . . . 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 7236 | . . . . 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 10825 | . . . . . 6 ⊢ 0 ∈ ℂ | |
9 | 8 | elimel 4508 | . . . . 5 ⊢ if(𝑦 ∈ ℂ, 𝑦, 0) ∈ ℂ |
10 | 4, 5, 6, 7, 9 | lnopunilem1 30091 | . . . 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 4497 | . . 3 ⊢ (𝑦 ∈ ℂ → (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵)))) |
12 | 11 | rgen 3071 | . 2 ⊢ ∀𝑦 ∈ ℂ (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) |
13 | 4 | lnopfi 30050 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ |
14 | 13 | ffvelrni 6903 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
15 | 6, 14 | ax-mp 5 | . . . 4 ⊢ (𝑇‘𝐴) ∈ ℋ |
16 | 13 | ffvelrni 6903 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℋ) |
17 | 7, 16 | ax-mp 5 | . . . 4 ⊢ (𝑇‘𝐵) ∈ ℋ |
18 | 15, 17 | hicli 29162 | . . 3 ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) ∈ ℂ |
19 | 6, 7 | hicli 29162 | . . 3 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
20 | recan 14900 | . . 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 692 | . 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 233 | 1 ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1543 ∈ wcel 2110 ∀wral 3061 ifcif 4439 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 0cc0 10729 · cmul 10734 ℜcre 14660 ℋchba 29000 ·ih csp 29003 normℎcno 29004 LinOpclo 29028 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 ax-hilex 29080 ax-hfvadd 29081 ax-hv0cl 29084 ax-hfvmul 29086 ax-hvmul0 29091 ax-hfi 29160 ax-his1 29163 ax-his2 29164 ax-his3 29165 ax-his4 29166 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-sup 9058 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-seq 13575 df-exp 13636 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-hnorm 29049 df-lnop 29922 |
This theorem is referenced by: lnopunii 30093 |
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