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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 20144 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 10613 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
9 | 5, 6, 7, 8 | syl3an 1157 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
10 | 0z 11980 | . . 3 ⊢ 0 ∈ ℤ | |
11 | 5 | addid2d 10830 | . . 3 ⊢ (𝑥 ∈ ℤ → (0 + 𝑥) = 𝑥) |
12 | znegcl 12005 | . . 3 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
13 | zcn 11974 | . . . . . 6 ⊢ (-𝑥 ∈ ℤ → -𝑥 ∈ ℂ) | |
14 | addcom 10815 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
15 | 5, 13, 14 | syl2an 598 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ -𝑥 ∈ ℤ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
16 | 12, 15 | mpdan 686 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
17 | 5 | negidd 10976 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = 0) |
18 | 16, 17 | eqtr3d 2835 | . . 3 ⊢ (𝑥 ∈ ℤ → (-𝑥 + 𝑥) = 0) |
19 | 1, 2, 3, 4, 9, 10, 11, 12, 18 | isgrpix 18122 | . 2 ⊢ 𝐺 ∈ Grp |
20 | 1, 2, 3 | grpbasex 16605 | . 2 ⊢ ℤ = (Base‘𝐺) |
21 | 1, 2, 3 | grpplusgx 16606 | . 2 ⊢ + = (+g‘𝐺) |
22 | addcom 10815 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
23 | 5, 6, 22 | syl2an 598 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
24 | 19, 20, 21, 23 | isabli 18913 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 {cpr 4527 〈cop 4531 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 -cneg 10860 2c2 11680 ℤcz 11969 Abelcabl 18899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-addf 10605 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-cmn 18900 df-abl 18901 |
This theorem is referenced by: (None) |
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