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| Mirrors > Home > HSE Home > Th. List > homullid | Structured version Visualization version GIF version | ||
| Description: An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| homullid | ⊢ (𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax-1cn 11213 | . . . . 5 ⊢ 1 ∈ ℂ | |
| 2 | homval 31760 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((1 ·op 𝑇)‘𝑥) = (1 ·ℎ (𝑇‘𝑥))) | |
| 3 | 1, 2 | mp3an1 1450 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((1 ·op 𝑇)‘𝑥) = (1 ·ℎ (𝑇‘𝑥))) | 
| 4 | ffvelcdm 7101 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ) | |
| 5 | ax-hvmulid 31025 | . . . . 5 ⊢ ((𝑇‘𝑥) ∈ ℋ → (1 ·ℎ (𝑇‘𝑥)) = (𝑇‘𝑥)) | |
| 6 | 4, 5 | syl 17 | . . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (1 ·ℎ (𝑇‘𝑥)) = (𝑇‘𝑥)) | 
| 7 | 3, 6 | eqtrd 2777 | . . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((1 ·op 𝑇)‘𝑥) = (𝑇‘𝑥)) | 
| 8 | 7 | ralrimiva 3146 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → ∀𝑥 ∈ ℋ ((1 ·op 𝑇)‘𝑥) = (𝑇‘𝑥)) | 
| 9 | homulcl 31778 | . . . 4 ⊢ ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op 𝑇): ℋ⟶ ℋ) | |
| 10 | 1, 9 | mpan 690 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → (1 ·op 𝑇): ℋ⟶ ℋ) | 
| 11 | hoeq 31779 | . . 3 ⊢ (((1 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ((1 ·op 𝑇)‘𝑥) = (𝑇‘𝑥) ↔ (1 ·op 𝑇) = 𝑇)) | |
| 12 | 10, 11 | mpancom 688 | . 2 ⊢ (𝑇: ℋ⟶ ℋ → (∀𝑥 ∈ ℋ ((1 ·op 𝑇)‘𝑥) = (𝑇‘𝑥) ↔ (1 ·op 𝑇) = 𝑇)) | 
| 13 | 8, 12 | mpbid 232 | 1 ⊢ (𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 1c1 11156 ℋchba 30938 ·ℎ csm 30940 ·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-hilex 31018 ax-hfvmul 31024 ax-hvmulid 31025 | 
| 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-homul 31750 | 
| This theorem is referenced by: honegneg 31825 ho2times 31838 leopmul 32153 nmopleid 32158 opsqrlem1 32159 opsqrlem6 32164 | 
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