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Mirrors > Home > MPE Home > Th. List > isgrpix | Structured version Visualization version GIF version |
Description: Properties that determine a group. Read 𝑁 as 𝑁(𝑥). Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. (New usage is discouraged.) (Contributed by NM, 4-Sep-2011.) |
Ref | Expression |
---|---|
isgrpix.a | ⊢ 𝐵 ∈ V |
isgrpix.b | ⊢ + ∈ V |
isgrpix.g | ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩} |
isgrpix.2 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
isgrpix.3 | ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) |
isgrpix.z | ⊢ 0 ∈ 𝐵 |
isgrpix.5 | ⊢ (𝑥 ∈ 𝐵 → ( 0 + 𝑥) = 𝑥) |
isgrpix.6 | ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵) |
isgrpix.7 | ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 ) |
Ref | Expression |
---|---|
isgrpix | ⊢ 𝐺 ∈ Grp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgrpix.a | . . 3 ⊢ 𝐵 ∈ V | |
2 | isgrpix.b | . . 3 ⊢ + ∈ V | |
3 | isgrpix.g | . . 3 ⊢ 𝐺 = {⟨1, 𝐵⟩, ⟨2, + ⟩} | |
4 | 1, 2, 3 | grpbasex 17243 | . 2 ⊢ 𝐵 = (Base‘𝐺) |
5 | 1, 2, 3 | grpplusgx 17244 | . 2 ⊢ + = (+g‘𝐺) |
6 | isgrpix.2 | . 2 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) | |
7 | isgrpix.3 | . 2 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
8 | isgrpix.z | . 2 ⊢ 0 ∈ 𝐵 | |
9 | isgrpix.5 | . 2 ⊢ (𝑥 ∈ 𝐵 → ( 0 + 𝑥) = 𝑥) | |
10 | isgrpix.6 | . 2 ⊢ (𝑥 ∈ 𝐵 → 𝑁 ∈ 𝐵) | |
11 | isgrpix.7 | . 2 ⊢ (𝑥 ∈ 𝐵 → (𝑁 + 𝑥) = 0 ) | |
12 | 4, 5, 6, 7, 8, 9, 10, 11 | isgrpi 18887 | 1 ⊢ 𝐺 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3468 {cpr 4625 ⟨cop 4629 (class class class)co 7404 1c1 11110 2c2 12268 Grpcgrp 18861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-0g 17394 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-grp 18864 |
This theorem is referenced by: cnaddablx 19786 zaddablx 19790 |
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