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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 20661 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 12338 | . . 3 ⊢ ℤ ∈ V | |
2 | addex 12738 | . . 3 ⊢ + ∈ V | |
3 | zaddablx.g | . . 3 ⊢ 𝐺 = {〈1, ℤ〉, 〈2, + 〉} | |
4 | zaddcl 12370 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) ∈ ℤ) | |
5 | zcn 12334 | . . . 4 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
6 | zcn 12334 | . . . 4 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ ℂ) | |
7 | zcn 12334 | . . . 4 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ) | |
8 | addass 10968 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
9 | 5, 6, 7, 8 | syl3an 1159 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
10 | 0z 12340 | . . 3 ⊢ 0 ∈ ℤ | |
11 | 5 | addid2d 11186 | . . 3 ⊢ (𝑥 ∈ ℤ → (0 + 𝑥) = 𝑥) |
12 | znegcl 12365 | . . 3 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
13 | zcn 12334 | . . . . . 6 ⊢ (-𝑥 ∈ ℤ → -𝑥 ∈ ℂ) | |
14 | addcom 11171 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ -𝑥 ∈ ℂ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) | |
15 | 5, 13, 14 | syl2an 596 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ -𝑥 ∈ ℤ) → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
16 | 12, 15 | mpdan 684 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = (-𝑥 + 𝑥)) |
17 | 5 | negidd 11332 | . . . 4 ⊢ (𝑥 ∈ ℤ → (𝑥 + -𝑥) = 0) |
18 | 16, 17 | eqtr3d 2780 | . . 3 ⊢ (𝑥 ∈ ℤ → (-𝑥 + 𝑥) = 0) |
19 | 1, 2, 3, 4, 9, 10, 11, 12, 18 | isgrpix 18616 | . 2 ⊢ 𝐺 ∈ Grp |
20 | 1, 2, 3 | grpbasex 17011 | . 2 ⊢ ℤ = (Base‘𝐺) |
21 | 1, 2, 3 | grpplusgx 17012 | . 2 ⊢ + = (+g‘𝐺) |
22 | addcom 11171 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
23 | 5, 6, 22 | syl2an 596 | . 2 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
24 | 19, 20, 21, 23 | isabli 19411 | 1 ⊢ 𝐺 ∈ Abel |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 {cpr 4563 〈cop 4567 (class class class)co 7267 ℂcc 10879 0cc0 10881 1c1 10882 + caddc 10884 -cneg 11216 2c2 12038 ℤcz 12329 Abelcabl 19397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-addf 10960 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-n0 12244 df-z 12330 df-uz 12593 df-fz 13250 df-struct 16858 df-slot 16893 df-ndx 16905 df-base 16923 df-plusg 16985 df-0g 17162 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-grp 18590 df-cmn 19398 df-abl 19399 |
This theorem is referenced by: (None) |
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