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Mirrors > Home > MPE Home > Th. List > zaddablx | Structured version Visualization version GIF version |
Description: The integers are an Abelian group under addition. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. Use zsubrg 19799 instead. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.) |
Ref | Expression |
---|---|
zaddablx.g | ⊢ 𝐺 = {〈1, ℤ〉, 〈2, + 〉} |
Ref | Expression |
---|---|
zaddablx | ⊢ 𝐺 ∈ Abel |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zex 11386 | . . 3 ⊢ ℤ ∈ V | |
2 | addex 11830 | . . 3 ⊢ + ∈ V | |
3 | zaddablx.g | . . 3 ⊢ 𝐺 = {〈1, ℤ〉, 〈2, + 〉} | |
4 | zaddcl 11417 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
5 | zcn 11382 | . . . 4 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
6 | zcn 11382 | . . . 4 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
7 | zcn 11382 | . . . 4 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ) | |
8 | addass 10023 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
9 | 5, 6, 7, 8 | syl3an 1368 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
10 | 0z 11388 | . . 3 ⊢ 0 ∈ ℤ | |
11 | 5 | addid2d 10237 | . . 3 ⊢ (𝑥 ∈ ℤ → (0 + 𝑥) = 𝑥) |
12 | znegcl 11412 | . . 3 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
13 | zcn 11382 | . . . . . 6 ⊢ (-𝑥 ∈ ℤ → -𝑥 ∈ ℂ) | |
14 | addcom 10222 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
15 | 5, 13, 14 | syl2an 494 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ -𝑥 ∈ ℤ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
16 | 12, 15 | mpdan 702 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
17 | 5 | negidd 10382 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = 0) |
18 | 16, 17 | eqtr3d 2658 | . . 3 ⊢ (𝑥 ∈ ℤ → (-𝑥 + 𝑥) = 0) |
19 | 1, 2, 3, 4, 9, 10, 11, 12, 18 | isgrpix 17449 | . 2 ⊢ 𝐺 ∈ Grp |
20 | 1, 2, 3 | grpbasex 15994 | . 2 ⊢ ℤ = (Base‘𝐺) |
21 | 1, 2, 3 | grpplusgx 15995 | . 2 ⊢ + = (+g‘𝐺) |
22 | addcom 10222 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
23 | 5, 6, 22 | syl2an 494 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
24 | 19, 20, 21, 23 | isabli 18207 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 {cpr 4179 〈cop 4183 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 -cneg 10267 2c2 11070 ℤcz 11377 Abelcabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: (None) |
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