frgpnabl - Metamath Proof Explorer

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Theorem frgpnabl 18278

Description: The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)

**Hypothesis**
Ref	Expression
frgpnabl.g	⊢ 𝐺 = (freeGrp‘𝐼)

**Assertion**
Ref	Expression
frgpnabl	⊢ (1_𝑜 ≺ 𝐼 → ¬ 𝐺 ∈ Abel)

Proof of Theorem frgpnabl Dummy variables 𝑎 𝑏 𝑥 𝑛 𝑣 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relsdom 7962	. . . . 5 ⊢ Rel ≺
2	1	brrelex2i 5159	. . . 4 ⊢ (1_𝑜 ≺ 𝐼 → 𝐼 ∈ V)
3		1sdom 8163	. . . 4 ⊢ (𝐼 ∈ V → (1_𝑜 ≺ 𝐼 ↔ ∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 𝐼 ¬ 𝑎 = 𝑏))
4	2, 3	syl 17	. . 3 ⊢ (1_𝑜 ≺ 𝐼 → (1_𝑜 ≺ 𝐼 ↔ ∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 𝐼 ¬ 𝑎 = 𝑏))
5	4	ibi 256	. 2 ⊢ (1_𝑜 ≺ 𝐼 → ∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 𝐼 ¬ 𝑎 = 𝑏)
6		frgpnabl.g	. . . . . 6 ⊢ 𝐺 = (freeGrp‘𝐼)
7		eqid 2622	. . . . . 6 ⊢ ( I ‘Word (𝐼 × 2_𝑜)) = ( I ‘Word (𝐼 × 2_𝑜))
8		eqid 2622	. . . . . 6 ⊢ ( ~_FG ‘𝐼) = ( ~_FG ‘𝐼)
9		eqid 2622	. . . . . 6 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
10		eqid 2622	. . . . . 6 ⊢ (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩) = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)
11		eqid 2622	. . . . . 6 ⊢ (𝑣 ∈ ( I ‘Word (𝐼 × 2_𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)‘𝑤)”⟩⟩))) = (𝑣 ∈ ( I ‘Word (𝐼 × 2_𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))
12		eqid 2622	. . . . . 6 ⊢ (( I ‘Word (𝐼 × 2_𝑜)) ∖ ∪ 𝑥 ∈ ( I ‘Word (𝐼 × 2_𝑜))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2_𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥)) = (( I ‘Word (𝐼 × 2_𝑜)) ∖ ∪ 𝑥 ∈ ( I ‘Word (𝐼 × 2_𝑜))ran ((𝑣 ∈ ( I ‘Word (𝐼 × 2_𝑜)) ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2_𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦 ∈ 𝐼, 𝑧 ∈ 2_𝑜 ↦ ⟨𝑦, (1_𝑜 ∖ 𝑧)⟩)‘𝑤)”⟩⟩)))‘𝑥))
13		eqid 2622	. . . . . 6 ⊢ (var_FGrp‘𝐼) = (var_FGrp‘𝐼)
14	2	ad2antrr 762	. . . . . 6 ⊢ (((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) ∧ 𝐺 ∈ Abel) → 𝐼 ∈ V)
15		simplrl 800	. . . . . 6 ⊢ (((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) ∧ 𝐺 ∈ Abel) → 𝑎 ∈ 𝐼)
16		simplrr 801	. . . . . 6 ⊢ (((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) ∧ 𝐺 ∈ Abel) → 𝑏 ∈ 𝐼)
17		simpr 477	. . . . . . 7 ⊢ (((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) ∧ 𝐺 ∈ Abel) → 𝐺 ∈ Abel)
18		eqid 2622	. . . . . . . . . 10 ⊢ (Base‘𝐺) = (Base‘𝐺)
19	8, 13, 6, 18	vrgpf 18181	. . . . . . . . 9 ⊢ (𝐼 ∈ V → (var_FGrp‘𝐼):𝐼⟶(Base‘𝐺))
20	14, 19	syl 17	. . . . . . . 8 ⊢ (((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) ∧ 𝐺 ∈ Abel) → (var_FGrp‘𝐼):𝐼⟶(Base‘𝐺))
21	20, 15	ffvelrnd 6360	. . . . . . 7 ⊢ (((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) ∧ 𝐺 ∈ Abel) → ((var_FGrp‘𝐼)‘𝑎) ∈ (Base‘𝐺))
22	20, 16	ffvelrnd 6360	. . . . . . 7 ⊢ (((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) ∧ 𝐺 ∈ Abel) → ((var_FGrp‘𝐼)‘𝑏) ∈ (Base‘𝐺))
23	18, 9	ablcom 18210	. . . . . . 7 ⊢ ((𝐺 ∈ Abel ∧ ((var_FGrp‘𝐼)‘𝑎) ∈ (Base‘𝐺) ∧ ((var_FGrp‘𝐼)‘𝑏) ∈ (Base‘𝐺)) → (((var_FGrp‘𝐼)‘𝑎)(+_g‘𝐺)((var_FGrp‘𝐼)‘𝑏)) = (((var_FGrp‘𝐼)‘𝑏)(+_g‘𝐺)((var_FGrp‘𝐼)‘𝑎)))
24	17, 21, 22, 23	syl3anc 1326	. . . . . 6 ⊢ (((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) ∧ 𝐺 ∈ Abel) → (((var_FGrp‘𝐼)‘𝑎)(+_g‘𝐺)((var_FGrp‘𝐼)‘𝑏)) = (((var_FGrp‘𝐼)‘𝑏)(+_g‘𝐺)((var_FGrp‘𝐼)‘𝑎)))
25	6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 24	frgpnabllem2 18277	. . . . 5 ⊢ (((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) ∧ 𝐺 ∈ Abel) → 𝑎 = 𝑏)
26	25	ex 450	. . . 4 ⊢ ((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) → (𝐺 ∈ Abel → 𝑎 = 𝑏))
27	26	con3d 148	. . 3 ⊢ ((1_𝑜 ≺ 𝐼 ∧ (𝑎 ∈ 𝐼 ∧ 𝑏 ∈ 𝐼)) → (¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ Abel))
28	27	rexlimdvva 3038	. 2 ⊢ (1_𝑜 ≺ 𝐼 → (∃𝑎 ∈ 𝐼 ∃𝑏 ∈ 𝐼 ¬ 𝑎 = 𝑏 → ¬ 𝐺 ∈ Abel))
29	5, 28	mpd 15	1 ⊢ (1_𝑜 ≺ 𝐼 → ¬ 𝐺 ∈ Abel)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 Vcvv 3200 ∖ cdif 3571 ⟨cop 4183 ⟨cotp 4185 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1_𝑜c1o 7553 2_𝑜c2o 7554 ≺ csdm 7954 0cc0 9936 ...cfz 12326 #chash 13117 Word cword 13291 splice csplice 13296 ⟨“cs2 13586 Basecbs 15857 +_gcplusg 15941 ~_FG cefg 18119 freeGrpcfrgp 18120 var_FGrpcvrgp 18121 Abelcabl 18194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-reverse 13305 df-s2 13593 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-0g 16102 df-imas 16168 df-qus 16169 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-frmd 17386 df-grp 17425 df-efg 18122 df-frgp 18123 df-vrgp 18124 df-cmn 18195 df-abl 18196

This theorem is referenced by: frgpcyg 19922

W3C validator