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Mirrors > Home > MPE Home > Th. List > cnaddabloOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cnaddabl 19808. 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 11205 | . . 3 β’ β β V | |
2 | ax-addf 11203 | . . 3 β’ + :(β Γ β)βΆβ | |
3 | addass 11211 | . . 3 β’ ((π₯ β β β§ π¦ β β β§ π§ β β) β ((π₯ + π¦) + π§) = (π₯ + (π¦ + π§))) | |
4 | 0cn 11222 | . . 3 β’ 0 β β | |
5 | addlid 11413 | . . 3 β’ (π₯ β β β (0 + π₯) = π₯) | |
6 | negcl 11476 | . . 3 β’ (π₯ β β β -π₯ β β) | |
7 | addcom 11416 | . . . . 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 11523 | . . . 4 β’ (π₯ β β β (π₯ + -π₯) = 0) | |
10 | 8, 9 | eqtr3d 2769 | . . 3 β’ (π₯ β β β (-π₯ + π₯) = 0) |
11 | 1, 2, 3, 4, 5, 6, 10 | isgrpoi 30282 | . 2 β’ + β GrpOp |
12 | 2 | fdmi 6728 | . 2 β’ dom + = (β Γ β) |
13 | addcom 11416 | . 2 β’ ((π₯ β β β§ π¦ β β) β (π₯ + π¦) = (π¦ + π₯)) | |
14 | 11, 12, 13 | isabloi 30335 | 1 β’ + β AbelOp |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 β wcel 2099 Γ cxp 5670 (class class class)co 7414 βcc 11122 0cc0 11124 + caddc 11127 -cneg 11461 AbelOpcablo 30328 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-addf 11203 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-ltxr 11269 df-sub 11462 df-neg 11463 df-grpo 30277 df-ablo 30329 |
This theorem is referenced by: cnidOLD 30366 cncvcOLD 30367 cnnv 30461 cnnvba 30463 cncph 30603 |
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