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Mirrors > Home > MPE Home > Th. List > cnaddabloOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cnaddabl 18983. Complex number addition is an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
cnaddabloOLD | ⊢ + ∈ AbelOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10612 | . . 3 ⊢ ℂ ∈ V | |
2 | ax-addf 10610 | . . 3 ⊢ + :(ℂ × ℂ)⟶ℂ | |
3 | addass 10618 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
4 | 0cn 10627 | . . 3 ⊢ 0 ∈ ℂ | |
5 | addid2 10817 | . . 3 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
6 | negcl 10880 | . . 3 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
7 | addcom 10820 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
8 | 6, 7 | mpdan 685 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
9 | negid 10927 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
10 | 8, 9 | eqtr3d 2858 | . . 3 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = 0) |
11 | 1, 2, 3, 4, 5, 6, 10 | isgrpoi 28269 | . 2 ⊢ + ∈ GrpOp |
12 | 2 | fdmi 6518 | . 2 ⊢ dom + = (ℂ × ℂ) |
13 | addcom 10820 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
14 | 11, 12, 13 | isabloi 28322 | 1 ⊢ + ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 × cxp 5547 (class class class)co 7150 ℂcc 10529 0cc0 10531 + caddc 10534 -cneg 10865 AbelOpcablo 28315 |
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-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-addf 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-sub 10866 df-neg 10867 df-grpo 28264 df-ablo 28316 |
This theorem is referenced by: cnidOLD 28353 cncvcOLD 28354 cnnv 28448 cnnvba 28450 cncph 28590 |
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