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Mirrors > Home > HSE Home > Th. List > ho2times | Structured version Visualization version GIF version |
Description: Two times a Hilbert space operator. (Contributed by NM, 26-Aug-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ho2times | ⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-2 11421 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 6920 | . . 3 ⊢ (2 ·op 𝑇) = ((1 + 1) ·op 𝑇) |
3 | ax-1cn 10317 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | hoadddir 29214 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) | |
5 | 3, 3, 4 | mp3an12 1579 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
6 | 2, 5 | syl5eq 2873 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
7 | hoadddi 29213 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) | |
8 | 3, 7 | mp3an1 1576 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
9 | 8 | anidms 562 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
10 | hoaddcl 29168 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑇 +op 𝑇): ℋ⟶ ℋ) | |
11 | 10 | anidms 562 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 +op 𝑇): ℋ⟶ ℋ) |
12 | homulid2 29210 | . . 3 ⊢ ((𝑇 +op 𝑇): ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = (𝑇 +op 𝑇)) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = (𝑇 +op 𝑇)) |
14 | 6, 9, 13 | 3eqtr2d 2867 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ⟶wf 6123 (class class class)co 6910 ℂcc 10257 1c1 10260 + caddc 10262 2c2 11413 ℋchba 28327 +op chos 28346 ·op chot 28347 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-1cn 10317 ax-addcl 10319 ax-hilex 28407 ax-hfvadd 28408 ax-hfvmul 28413 ax-hvmulid 28414 ax-hvdistr1 28416 ax-hvdistr2 28417 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-2 11421 df-hosum 29140 df-homul 29141 |
This theorem is referenced by: opsqrlem6 29555 |
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