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Mirrors > Home > MPE Home > Th. List > cncvcOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cncvs 24454. The set of complex numbers is a complex vector space. The vector operation is +, and the scalar product is ·. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
cncvcOLD | ⊢ 〈 + , · 〉 ∈ CVecOLD |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnaddabloOLD 29368 | . 2 ⊢ + ∈ AbelOp | |
2 | ax-addf 11088 | . . 3 ⊢ + :(ℂ × ℂ)⟶ℂ | |
3 | 2 | fdmi 6677 | . 2 ⊢ dom + = (ℂ × ℂ) |
4 | ax-mulf 11089 | . 2 ⊢ · :(ℂ × ℂ)⟶ℂ | |
5 | mulid2 11112 | . 2 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
6 | adddi 11098 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧))) | |
7 | adddir 11104 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 + 𝑧) · 𝑥) = ((𝑦 · 𝑥) + (𝑧 · 𝑥))) | |
8 | mulass 11097 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 · 𝑧) · 𝑥) = (𝑦 · (𝑧 · 𝑥))) | |
9 | eqid 2736 | . 2 ⊢ 〈 + , · 〉 = 〈 + , · 〉 | |
10 | 1, 3, 4, 5, 6, 7, 8, 9 | isvciOLD 29367 | 1 ⊢ 〈 + , · 〉 ∈ CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 〈cop 4590 × cxp 5629 ℂcc 11007 + caddc 11012 · cmul 11014 CVecOLDcvc 29345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-addf 11088 ax-mulf 11089 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-sub 11345 df-neg 11346 df-grpo 29280 df-ablo 29332 df-vc 29346 |
This theorem is referenced by: cnnv 29464 |
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