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Mirrors > Home > HSE Home > Th. List > unopnorm | Structured version Visualization version GIF version |
Description: A unitary operator is idempotent in the norm. (Contributed by NM, 25-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
unopnorm | ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) = (normℎ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unopf1o 29851 | . . . . 5 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ–1-1-onto→ ℋ) | |
2 | f1of 6618 | . . . . 5 ⊢ (𝑇: ℋ–1-1-onto→ ℋ → 𝑇: ℋ⟶ ℋ) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑇 ∈ UniOp → 𝑇: ℋ⟶ ℋ) |
4 | 3 | ffvelrnda 6861 | . . 3 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (𝑇‘𝐴) ∈ ℋ) |
5 | normcl 29060 | . . 3 ⊢ ((𝑇‘𝐴) ∈ ℋ → (normℎ‘(𝑇‘𝐴)) ∈ ℝ) | |
6 | 4, 5 | syl 17 | . 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) ∈ ℝ) |
7 | normcl 29060 | . . 3 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) | |
8 | 7 | adantl 485 | . 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (normℎ‘𝐴) ∈ ℝ) |
9 | normge0 29061 | . . 3 ⊢ ((𝑇‘𝐴) ∈ ℋ → 0 ≤ (normℎ‘(𝑇‘𝐴))) | |
10 | 4, 9 | syl 17 | . 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → 0 ≤ (normℎ‘(𝑇‘𝐴))) |
11 | normge0 29061 | . . 3 ⊢ (𝐴 ∈ ℋ → 0 ≤ (normℎ‘𝐴)) | |
12 | 11 | adantl 485 | . 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → 0 ≤ (normℎ‘𝐴)) |
13 | unop 29850 | . . . 4 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·ih (𝑇‘𝐴)) = (𝐴 ·ih 𝐴)) | |
14 | 13 | 3anidm23 1422 | . . 3 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → ((𝑇‘𝐴) ·ih (𝑇‘𝐴)) = (𝐴 ·ih 𝐴)) |
15 | normsq 29069 | . . . 4 ⊢ ((𝑇‘𝐴) ∈ ℋ → ((normℎ‘(𝑇‘𝐴))↑2) = ((𝑇‘𝐴) ·ih (𝑇‘𝐴))) | |
16 | 4, 15 | syl 17 | . . 3 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → ((normℎ‘(𝑇‘𝐴))↑2) = ((𝑇‘𝐴) ·ih (𝑇‘𝐴))) |
17 | normsq 29069 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴)) | |
18 | 17 | adantl 485 | . . 3 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → ((normℎ‘𝐴)↑2) = (𝐴 ·ih 𝐴)) |
19 | 14, 16, 18 | 3eqtr4d 2783 | . 2 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → ((normℎ‘(𝑇‘𝐴))↑2) = ((normℎ‘𝐴)↑2)) |
20 | 6, 8, 10, 12, 19 | sq11d 13713 | 1 ⊢ ((𝑇 ∈ UniOp ∧ 𝐴 ∈ ℋ) → (normℎ‘(𝑇‘𝐴)) = (normℎ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 class class class wbr 5030 ⟶wf 6335 –1-1-onto→wf1o 6338 ‘cfv 6339 (class class class)co 7170 ℝcr 10614 0cc0 10615 ≤ cle 10754 2c2 11771 ↑cexp 13521 ℋchba 28854 ·ih csp 28857 normℎcno 28858 UniOpcuo 28884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-hilex 28934 ax-hfvadd 28935 ax-hvcom 28936 ax-hvass 28937 ax-hv0cl 28938 ax-hvaddid 28939 ax-hfvmul 28940 ax-hvmulid 28941 ax-hvdistr2 28944 ax-hvmul0 28945 ax-hfi 29014 ax-his1 29017 ax-his2 29018 ax-his3 29019 ax-his4 29020 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-hnorm 28903 df-hvsub 28906 df-unop 29778 |
This theorem is referenced by: elunop2 29948 nmopun 29949 |
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