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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 12329 | . . . 4 ⊢ 2 = (1 + 1) | |
| 2 | 1 | oveq1i 7441 | . . 3 ⊢ (2 ·op 𝑇) = ((1 + 1) ·op 𝑇) |
| 3 | ax-1cn 11213 | . . . 4 ⊢ 1 ∈ ℂ | |
| 4 | hoadddir 31823 | . . . 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 | eqtrid 2789 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
| 7 | hoadddi 31822 | . . . 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 566 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (1 ·op (𝑇 +op 𝑇)) = ((1 ·op 𝑇) +op (1 ·op 𝑇))) |
| 10 | hoaddcl 31777 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑇 +op 𝑇): ℋ⟶ ℋ) | |
| 11 | 10 | anidms 566 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (𝑇 +op 𝑇): ℋ⟶ ℋ) |
| 12 | homullid 31819 | . . 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 2783 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (2 ·op 𝑇) = (𝑇 +op 𝑇)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⟶wf 6557 (class class class)co 7431 ℂcc 11153 1c1 11156 + caddc 11158 2c2 12321 ℋchba 30938 +op chos 30957 ·op chot 30958 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-1cn 11213 ax-addcl 11215 ax-hilex 31018 ax-hfvadd 31019 ax-hfvmul 31024 ax-hvmulid 31025 ax-hvdistr1 31027 ax-hvdistr2 31028 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-map 8868 df-2 12329 df-hosum 31749 df-homul 31750 |
| This theorem is referenced by: opsqrlem6 32164 |
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