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Mirrors > Home > MPE Home > Th. List > cncvcOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cncvs 25192. 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 30610 | . 2 ⊢ + ∈ AbelOp | |
2 | ax-addf 11232 | . . 3 ⊢ + :(ℂ × ℂ)⟶ℂ | |
3 | 2 | fdmi 6748 | . 2 ⊢ dom + = (ℂ × ℂ) |
4 | ax-mulf 11233 | . 2 ⊢ · :(ℂ × ℂ)⟶ℂ | |
5 | mullid 11258 | . 2 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
6 | adddi 11242 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧))) | |
7 | adddir 11250 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 + 𝑧) · 𝑥) = ((𝑦 · 𝑥) + (𝑧 · 𝑥))) | |
8 | mulass 11241 | . 2 ⊢ ((𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → ((𝑦 · 𝑧) · 𝑥) = (𝑦 · (𝑧 · 𝑥))) | |
9 | eqid 2735 | . 2 ⊢ 〈 + , · 〉 = 〈 + , · 〉 | |
10 | 1, 3, 4, 5, 6, 7, 8, 9 | isvciOLD 30609 | 1 ⊢ 〈 + , · 〉 ∈ CVecOLD |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 〈cop 4637 × cxp 5687 ℂcc 11151 + caddc 11156 · cmul 11158 CVecOLDcvc 30587 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-grpo 30522 df-ablo 30574 df-vc 30588 |
This theorem is referenced by: cnnv 30706 |
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