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Mirrors > Home > HSE Home > Th. List > lnopunilem2 | Structured version Visualization version GIF version |
Description: Lemma for lnopunii 28255. (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 | oveq1 6556 | . . . . . 6 ⊢ (𝑦 = if(𝑦 ∈ ℂ, 𝑦, 0) → (𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵))) = (if(𝑦 ∈ ℂ, 𝑦, 0) · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) | |
2 | 1 | fveq2d 6107 | . . . . 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))))) |
3 | oveq1 6556 | . . . . . 6 ⊢ (𝑦 = if(𝑦 ∈ ℂ, 𝑦, 0) → (𝑦 · (𝐴 ·ih 𝐵)) = (if(𝑦 ∈ ℂ, 𝑦, 0) · (𝐴 ·ih 𝐵))) | |
4 | 3 | fveq2d 6107 | . . . . 5 ⊢ (𝑦 = if(𝑦 ∈ ℂ, 𝑦, 0) → (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) = (ℜ‘(if(𝑦 ∈ ℂ, 𝑦, 0) · (𝐴 ·ih 𝐵)))) |
5 | 2, 4 | eqeq12d 2625 | . . . 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))))) |
6 | lnopunilem.1 | . . . . 5 ⊢ 𝑇 ∈ LinOp | |
7 | lnopunilem.2 | . . . . 5 ⊢ ∀𝑥 ∈ ℋ (normℎ‘(𝑇‘𝑥)) = (normℎ‘𝑥) | |
8 | lnopunilem.3 | . . . . 5 ⊢ 𝐴 ∈ ℋ | |
9 | lnopunilem.4 | . . . . 5 ⊢ 𝐵 ∈ ℋ | |
10 | 0cn 9911 | . . . . . 6 ⊢ 0 ∈ ℂ | |
11 | 10 | elimel 4100 | . . . . 5 ⊢ if(𝑦 ∈ ℂ, 𝑦, 0) ∈ ℂ |
12 | 6, 7, 8, 9, 11 | lnopunilem1 28253 | . . . 4 ⊢ (ℜ‘(if(𝑦 ∈ ℂ, 𝑦, 0) · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(if(𝑦 ∈ ℂ, 𝑦, 0) · (𝐴 ·ih 𝐵))) |
13 | 5, 12 | dedth 4089 | . . 3 ⊢ (𝑦 ∈ ℂ → (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵)))) |
14 | 13 | rgen 2906 | . 2 ⊢ ∀𝑦 ∈ ℂ (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) |
15 | 6 | lnopfi 28212 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ |
16 | 15 | ffvelrni 6266 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
17 | 8, 16 | ax-mp 5 | . . . 4 ⊢ (𝑇‘𝐴) ∈ ℋ |
18 | 15 | ffvelrni 6266 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝑇‘𝐵) ∈ ℋ) |
19 | 9, 18 | ax-mp 5 | . . . 4 ⊢ (𝑇‘𝐵) ∈ ℋ |
20 | 17, 19 | hicli 27322 | . . 3 ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) ∈ ℂ |
21 | 8, 9 | hicli 27322 | . . 3 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
22 | recan 13924 | . . 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))) | |
23 | 20, 21, 22 | mp2an 704 | . 2 ⊢ (∀𝑦 ∈ ℂ (ℜ‘(𝑦 · ((𝑇‘𝐴) ·ih (𝑇‘𝐵)))) = (ℜ‘(𝑦 · (𝐴 ·ih 𝐵))) ↔ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵)) |
24 | 14, 23 | mpbi 219 | 1 ⊢ ((𝑇‘𝐴) ·ih (𝑇‘𝐵)) = (𝐴 ·ih 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ifcif 4036 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 · cmul 9820 ℜcre 13685 ℋchil 27160 ·ih csp 27163 normℎcno 27164 LinOpclo 27188 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-hilex 27240 ax-hfvadd 27241 ax-hv0cl 27244 ax-hfvmul 27246 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his2 27324 ax-his3 27325 ax-his4 27326 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-hnorm 27209 df-lnop 28084 |
This theorem is referenced by: lnopunii 28255 |
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