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Mirrors > Home > MPE Home > Th. List > cncvcOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cncvs 24661. 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 29834 | . 2 ⊢ + ∈ AbelOp | |
2 | ax-addf 11189 | . . 3 ⊢ + :(ℂ × ℂ)⟶ℂ | |
3 | 2 | fdmi 6730 | . 2 ⊢ dom + = (ℂ × ℂ) |
4 | ax-mulf 11190 | . 2 ⊢ · :(ℂ × ℂ)⟶ℂ | |
5 | mullid 11213 | . 2 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
6 | adddi 11199 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧))) | |
7 | adddir 11205 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 + 𝑧) · 𝑥) = ((𝑦 · 𝑥) + (𝑧 · 𝑥))) | |
8 | mulass 11198 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 · 𝑧) · 𝑥) = (𝑦 · (𝑧 · 𝑥))) | |
9 | eqid 2733 | . 2 ⊢ ⟨ + , · ⟩ = ⟨ + , · ⟩ | |
10 | 1, 3, 4, 5, 6, 7, 8, 9 | isvciOLD 29833 | 1 ⊢ ⟨ + , · ⟩ ∈ CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ⟨cop 4635 × cxp 5675 ℂcc 11108 + caddc 11113 · cmul 11115 CVecOLDcvc 29811 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-addf 11189 ax-mulf 11190 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-po 5589 df-so 5590 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-ltxr 11253 df-sub 11446 df-neg 11447 df-grpo 29746 df-ablo 29798 df-vc 29812 |
This theorem is referenced by: cnnv 29930 |
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