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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 17272 | . 2 ⊢ 𝐵 = (Base‘𝐺) |
5 | 1, 2, 3 | grpplusgx 17273 | . 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 18916 | 1 ⊢ 𝐺 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3471 {cpr 4631 〈cop 4635 (class class class)co 7420 1c1 11140 2c2 12298 Grpcgrp 18890 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-slot 17151 df-ndx 17163 df-base 17181 df-plusg 17246 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18893 |
This theorem is referenced by: cnaddablx 19823 zaddablx 19827 |
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