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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 11858 | . . . 4 ⊢ 2 = (1 + 1) | |
2 | 1 | oveq1i 7201 | . . 3 ⊢ (2 ·op 𝑇) = ((1 + 1) ·op 𝑇) |
3 | ax-1cn 10752 | . . . 4 ⊢ 1 ∈ ℂ | |
4 | hoadddir 29839 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) | |
5 | 3, 3, 4 | mp3an12 1453 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → ((1 + 1) ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
6 | 2, 5 | syl5eq 2783 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
7 | hoadddi 29838 | . . . 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 1450 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
9 | 8 | anidms 570 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
10 | hoaddcl 29793 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑇 +op 𝑇): ℋ⟶ ℋ) | |
11 | 10 | anidms 570 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 +op 𝑇): ℋ⟶ ℋ) |
12 | homulid2 29835 | . . 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 2777 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ⟶wf 6354 (class class class)co 7191 ℂcc 10692 1c1 10695 + caddc 10697 2c2 11850 ℋchba 28954 +op chos 28973 ·op chot 28974 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-1cn 10752 ax-addcl 10754 ax-hilex 29034 ax-hfvadd 29035 ax-hfvmul 29040 ax-hvmulid 29041 ax-hvdistr1 29043 ax-hvdistr2 29044 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-map 8488 df-2 11858 df-hosum 29765 df-homul 29766 |
This theorem is referenced by: opsqrlem6 30180 |
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