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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 20526 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 11978 | . . 3 ⊢ ℤ ∈ V | |
2 | addex 12375 | . . 3 ⊢ + ∈ V | |
3 | zaddablx.g | . . 3 ⊢ 𝐺 = {〈1, ℤ〉, 〈2, + 〉} | |
4 | zaddcl 12010 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
5 | zcn 11974 | . . . 4 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
6 | zcn 11974 | . . . 4 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
7 | zcn 11974 | . . . 4 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ) | |
8 | addass 10612 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
9 | 5, 6, 7, 8 | syl3an 1152 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
10 | 0z 11980 | . . 3 ⊢ 0 ∈ ℤ | |
11 | 5 | addid2d 10829 | . . 3 ⊢ (𝑥 ∈ ℤ → (0 + 𝑥) = 𝑥) |
12 | znegcl 12005 | . . 3 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
13 | zcn 11974 | . . . . . 6 ⊢ (-𝑥 ∈ ℤ → -𝑥 ∈ ℂ) | |
14 | addcom 10814 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
15 | 5, 13, 14 | syl2an 595 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ -𝑥 ∈ ℤ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
16 | 12, 15 | mpdan 683 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
17 | 5 | negidd 10975 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = 0) |
18 | 16, 17 | eqtr3d 2855 | . . 3 ⊢ (𝑥 ∈ ℤ → (-𝑥 + 𝑥) = 0) |
19 | 1, 2, 3, 4, 9, 10, 11, 12, 18 | isgrpix 18068 | . 2 ⊢ 𝐺 ∈ Grp |
20 | 1, 2, 3 | grpbasex 16601 | . 2 ⊢ ℤ = (Base‘𝐺) |
21 | 1, 2, 3 | grpplusgx 16602 | . 2 ⊢ + = (+g‘𝐺) |
22 | addcom 10814 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
23 | 5, 6, 22 | syl2an 595 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
24 | 19, 20, 21, 23 | isabli 18850 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 {cpr 4559 〈cop 4563 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 + caddc 10528 -cneg 10859 2c2 11680 ℤcz 11969 Abelcabl 18836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-addf 10604 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12881 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-plusg 16566 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-cmn 18837 df-abl 18838 |
This theorem is referenced by: (None) |
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