Theorem homulid2 29683

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.

Step	Hyp	Ref	Expression
1		ax-1cn 10634	. . . . 5 ⊢ 1 ∈ ℂ
2		homval 29624	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((1 ·op 𝑇)‘𝑥) = (1 ·ℎ (𝑇‘𝑥)))
3	1, 2	mp3an1 1446	. . . 4 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((1 ·op 𝑇)‘𝑥) = (1 ·ℎ (𝑇‘𝑥)))
4		ffvelrn 6841	. . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ)
5		ax-hvmulid 28889	. . . . 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 2794	. . 3 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((1 ·op 𝑇)‘𝑥) = (𝑇‘𝑥))
8	7	ralrimiva 3114	. 2 ⊢ (𝑇: ℋ⟶ ℋ → ∀𝑥 ∈ ℋ ((1 ·op 𝑇)‘𝑥) = (𝑇‘𝑥))
9		homulcl 29642	. . . 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 29643	. . 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 235	1 ⊢ (𝑇: ℋ⟶ ℋ → (1 ·op 𝑇) = 𝑇)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  ⟶wf 6332  ‘cfv 6336  (class class class)co 7151  ℂcc 10574  1c1 10577   ℋchba 28802   ·_ℎ csm 28804   ·_op chot 28822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-1cn 10634  ax-hilex 28882  ax-hfvmul 28888  ax-hvmulid 28889

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8419  df-homul 29614

This theorem is referenced by:  honegneg  29689  ho2times  29702  leopmul  30017  nmopleid  30022  opsqrlem1  30023  opsqrlem6  30028