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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 19618 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 11263 | . . 3 ⊢ ℤ ∈ V | |
2 | addex 11706 | . . 3 ⊢ + ∈ V | |
3 | zaddablx.g | . . 3 ⊢ 𝐺 = {〈1, ℤ〉, 〈2, + 〉} | |
4 | zaddcl 11294 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
5 | zcn 11259 | . . . 4 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
6 | zcn 11259 | . . . 4 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
7 | zcn 11259 | . . . 4 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ) | |
8 | addass 9902 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
9 | 5, 6, 7, 8 | syl3an 1360 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
10 | 0z 11265 | . . 3 ⊢ 0 ∈ ℤ | |
11 | 5 | addid2d 10116 | . . 3 ⊢ (𝑥 ∈ ℤ → (0 + 𝑥) = 𝑥) |
12 | znegcl 11289 | . . 3 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
13 | zcn 11259 | . . . . . 6 ⊢ (-𝑥 ∈ ℤ → -𝑥 ∈ ℂ) | |
14 | addcom 10101 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
15 | 5, 13, 14 | syl2an 493 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ -𝑥 ∈ ℤ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
16 | 12, 15 | mpdan 699 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
17 | 5 | negidd 10261 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = 0) |
18 | 16, 17 | eqtr3d 2646 | . . 3 ⊢ (𝑥 ∈ ℤ → (-𝑥 + 𝑥) = 0) |
19 | 1, 2, 3, 4, 9, 10, 11, 12, 18 | isgrpix 17272 | . 2 ⊢ 𝐺 ∈ Grp |
20 | 1, 2, 3 | grpbasex 15819 | . 2 ⊢ ℤ = (Base‘𝐺) |
21 | 1, 2, 3 | grpplusgx 15820 | . 2 ⊢ + = (+g‘𝐺) |
22 | addcom 10101 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
23 | 5, 6, 22 | syl2an 493 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
24 | 19, 20, 21, 23 | isabli 18030 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 {cpr 4127 〈cop 4131 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 -cneg 10146 2c2 10947 ℤcz 11254 Abelcabl 18017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-cmn 18018 df-abl 18019 |
This theorem is referenced by: (None) |
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