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Theorem homulid2 28989
Description: An operator equals its scalar product with one. (Contributed by NM, 12-Aug-2006.) (New usage is discouraged.)
Assertion
Ref Expression
homulid2 (𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇)

Proof of Theorem homulid2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ax-1cn 10206 . . . . 5 1 ∈ ℂ
2 homval 28930 . . . . 5 ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((1 ·op 𝑇)‘𝑥) = (1 · (𝑇𝑥)))
31, 2mp3an1 1560 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((1 ·op 𝑇)‘𝑥) = (1 · (𝑇𝑥)))
4 ffvelrn 6521 . . . . 5 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇𝑥) ∈ ℋ)
5 ax-hvmulid 28193 . . . . 5 ((𝑇𝑥) ∈ ℋ → (1 · (𝑇𝑥)) = (𝑇𝑥))
64, 5syl 17 . . . 4 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (1 · (𝑇𝑥)) = (𝑇𝑥))
73, 6eqtrd 2794 . . 3 ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((1 ·op 𝑇)‘𝑥) = (𝑇𝑥))
87ralrimiva 3104 . 2 (𝑇: ℋ⟶ ℋ → ∀𝑥 ∈ ℋ ((1 ·op 𝑇)‘𝑥) = (𝑇𝑥))
9 homulcl 28948 . . . 4 ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (1 ·op 𝑇): ℋ⟶ ℋ)
101, 9mpan 708 . . 3 (𝑇: ℋ⟶ ℋ → (1 ·op 𝑇): ℋ⟶ ℋ)
11 hoeq 28949 . . 3 (((1 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ((1 ·op 𝑇)‘𝑥) = (𝑇𝑥) ↔ (1 ·op 𝑇) = 𝑇))
1210, 11mpancom 706 . 2 (𝑇: ℋ⟶ ℋ → (∀𝑥 ∈ ℋ ((1 ·op 𝑇)‘𝑥) = (𝑇𝑥) ↔ (1 ·op 𝑇) = 𝑇))
138, 12mpbid 222 1 (𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  wf 6045  cfv 6049  (class class class)co 6814  cc 10146  1c1 10149  chil 28106   · csm 28108   ·op chot 28126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-1cn 10206  ax-hilex 28186  ax-hfvmul 28192  ax-hvmulid 28193
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-homul 28920
This theorem is referenced by:  honegneg  28995  ho2times  29008  leopmul  29323  nmopleid  29328  opsqrlem1  29329  opsqrlem6  29334
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