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Mirrors > Home > MPE Home > Th. List > vcidOLD | Structured version Visualization version GIF version |
Description: Identity element for the scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) Obsolete theorem, use clmvs1 23696 together with cvsclm 23729 instead. (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
vciOLD.1 | ⊢ 𝐺 = (1st ‘𝑊) |
vciOLD.2 | ⊢ 𝑆 = (2nd ‘𝑊) |
vciOLD.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
vcidOLD | ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vciOLD.1 | . . . 4 ⊢ 𝐺 = (1st ‘𝑊) | |
2 | vciOLD.2 | . . . 4 ⊢ 𝑆 = (2nd ‘𝑊) | |
3 | vciOLD.3 | . . . 4 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | vciOLD 28337 | . . 3 ⊢ (𝑊 ∈ CVecOLD → (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))))) |
5 | simpl 485 | . . . . 5 ⊢ (((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))) → (1𝑆𝑥) = 𝑥) | |
6 | 5 | ralimi 3160 | . . . 4 ⊢ (∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥))))) → ∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥) |
7 | 6 | 3ad2ant3 1131 | . . 3 ⊢ ((𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))) → ∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥) |
8 | 4, 7 | syl 17 | . 2 ⊢ (𝑊 ∈ CVecOLD → ∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥) |
9 | oveq2 7163 | . . . 4 ⊢ (𝑥 = 𝐴 → (1𝑆𝑥) = (1𝑆𝐴)) | |
10 | id 22 | . . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
11 | 9, 10 | eqeq12d 2837 | . . 3 ⊢ (𝑥 = 𝐴 → ((1𝑆𝑥) = 𝑥 ↔ (1𝑆𝐴) = 𝐴)) |
12 | 11 | rspccva 3621 | . 2 ⊢ ((∀𝑥 ∈ 𝑋 (1𝑆𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
13 | 8, 12 | sylan 582 | 1 ⊢ ((𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋) → (1𝑆𝐴) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 × cxp 5552 ran crn 5555 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 1st c1st 7686 2nd c2nd 7687 ℂcc 10534 1c1 10537 + caddc 10539 · cmul 10541 AbelOpcablo 28320 CVecOLDcvc 28334 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-ov 7158 df-1st 7688 df-2nd 7689 df-vc 28335 |
This theorem is referenced by: vc2OLD 28344 vc0 28350 vcm 28352 nvsid 28403 |
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