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Mirrors > Home > MPE Home > Th. List > cnaddabloOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cnaddabl 19655. 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 | β’ + β AbelOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 11140 | . . 3 β’ β β V | |
2 | ax-addf 11138 | . . 3 β’ + :(β Γ β)βΆβ | |
3 | addass 11146 | . . 3 β’ ((π₯ β β β§ π¦ β β β§ π§ β β) β ((π₯ + π¦) + π§) = (π₯ + (π¦ + π§))) | |
4 | 0cn 11155 | . . 3 β’ 0 β β | |
5 | addid2 11346 | . . 3 β’ (π₯ β β β (0 + π₯) = π₯) | |
6 | negcl 11409 | . . 3 β’ (π₯ β β β -π₯ β β) | |
7 | addcom 11349 | . . . . 5 β’ ((π₯ β β β§ -π₯ β β) β (π₯ + -π₯) = (-π₯ + π₯)) | |
8 | 6, 7 | mpdan 686 | . . . 4 β’ (π₯ β β β (π₯ + -π₯) = (-π₯ + π₯)) |
9 | negid 11456 | . . . 4 β’ (π₯ β β β (π₯ + -π₯) = 0) | |
10 | 8, 9 | eqtr3d 2775 | . . 3 β’ (π₯ β β β (-π₯ + π₯) = 0) |
11 | 1, 2, 3, 4, 5, 6, 10 | isgrpoi 29489 | . 2 β’ + β GrpOp |
12 | 2 | fdmi 6684 | . 2 β’ dom + = (β Γ β) |
13 | addcom 11349 | . 2 β’ ((π₯ β β β§ π¦ β β) β (π₯ + π¦) = (π¦ + π₯)) | |
14 | 11, 12, 13 | isabloi 29542 | 1 β’ + β AbelOp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 β wcel 2107 Γ cxp 5635 (class class class)co 7361 βcc 11057 0cc0 11059 + caddc 11062 -cneg 11394 AbelOpcablo 29535 |
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 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-addf 11138 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-po 5549 df-so 5550 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-ltxr 11202 df-sub 11395 df-neg 11396 df-grpo 29484 df-ablo 29536 |
This theorem is referenced by: cnidOLD 29573 cncvcOLD 29574 cnnv 29668 cnnvba 29670 cncph 29810 |
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