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Mirrors > Home > MPE Home > Th. List > cnaddabl | Structured version Visualization version GIF version |
Description: The complex numbers are an Abelian group under addition. This version of cnaddablx 18990 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how Base and +g is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnring 20569. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnaddabl.g | ⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} |
Ref | Expression |
---|---|
cnaddabl | ⊢ 𝐺 ∈ Abel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10620 | . . . 4 ⊢ ℂ ∈ V | |
2 | cnaddabl.g | . . . . 5 ⊢ 𝐺 = {〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉} | |
3 | 2 | grpbase 16612 | . . . 4 ⊢ (ℂ ∈ V → ℂ = (Base‘𝐺)) |
4 | 1, 3 | ax-mp 5 | . . 3 ⊢ ℂ = (Base‘𝐺) |
5 | addex 12390 | . . . 4 ⊢ + ∈ V | |
6 | 2 | grpplusg 16613 | . . . 4 ⊢ ( + ∈ V → + = (+g‘𝐺)) |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝐺) |
8 | addcl 10621 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
9 | addass 10626 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
10 | 0cn 10635 | . . 3 ⊢ 0 ∈ ℂ | |
11 | addid2 10825 | . . 3 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
12 | negcl 10888 | . . 3 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
13 | addcom 10828 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
14 | 12, 13 | mpdan 685 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
15 | negid 10935 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
16 | 14, 15 | eqtr3d 2860 | . . 3 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = 0) |
17 | 4, 7, 8, 9, 10, 11, 12, 16 | isgrpi 18128 | . 2 ⊢ 𝐺 ∈ Grp |
18 | addcom 10828 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
19 | 17, 4, 7, 18 | isabli 18923 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 {cpr 4571 〈cop 4575 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 0cc0 10539 + caddc 10542 -cneg 10873 ndxcnx 16482 Basecbs 16485 +gcplusg 16567 Abelcabl 18909 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-addf 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-plusg 16580 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-cmn 18910 df-abl 18911 |
This theorem is referenced by: cnaddinv 18993 cnaddcom 36110 |
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