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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 17177 | . 2 ⊢ 𝐵 = (Base‘𝐺) |
5 | 1, 2, 3 | grpplusgx 17178 | . 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 18778 | 1 ⊢ 𝐺 ∈ Grp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3444 {cpr 4589 ⟨cop 4593 (class class class)co 7358 1c1 11057 2c2 12213 Grpcgrp 18753 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-n0 12419 df-z 12505 df-uz 12769 df-fz 13431 df-struct 17024 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-0g 17328 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 |
This theorem is referenced by: cnaddablx 19651 zaddablx 19655 |
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