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Mirrors > Home > HSE Home > Th. List > lnopunilem2 | Structured version Visualization version GIF version |
Description: Lemma for lnopunii 29716. (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 7168 | . . . . 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 7168 | . . . . 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 2834 | . . . 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 10621 | . . . . . 6 ⊢ 0 ∈ ℂ | |
9 | 8 | elimel 4530 | . . . . 5 ⊢ if(𝑦 ∈ ℂ, 𝑦, 0) ∈ ℂ |
10 | 4, 5, 6, 7, 9 | lnopunilem1 29714 | . . . 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 4519 | . . 3 ⊢ (𝑦 ∈ ℂ → (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵)))) |
12 | 11 | rgen 3145 | . 2 ⊢ ∀𝑦 ∈ ℂ (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) |
13 | 4 | lnopfi 29673 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ |
14 | 13 | ffvelrni 6842 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
15 | 6, 14 | ax-mp 5 | . . . 4 ⊢ (𝑇‘𝐴) ∈ ℋ |
16 | 13 | ffvelrni 6842 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℋ) |
17 | 7, 16 | ax-mp 5 | . . . 4 ⊢ (𝑇‘𝐵) ∈ ℋ |
18 | 15, 17 | hicli 28785 | . . 3 ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) ∈ ℂ |
19 | 6, 7 | hicli 28785 | . . 3 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
20 | recan 14684 | . . 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 688 | . 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 231 | 1 ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ifcif 4463 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 · cmul 10530 ℜcre 14444 ℋchba 28623 ·ih csp 28626 normℎcno 28627 LinOpclo 28651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-hilex 28703 ax-hfvadd 28704 ax-hv0cl 28707 ax-hfvmul 28709 ax-hvmul0 28714 ax-hfi 28783 ax-his1 28786 ax-his2 28787 ax-his3 28788 ax-his4 28789 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-hnorm 28672 df-lnop 29545 |
This theorem is referenced by: lnopunii 29716 |
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