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Theorem frgp0 18373
 Description: The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
frgp0.m 𝐺 = (freeGrp‘𝐼)
frgp0.r = ( ~FG𝐼)
Assertion
Ref Expression
frgp0 (𝐼𝑉 → (𝐺 ∈ Grp ∧ [∅] = (0g𝐺)))

Proof of Theorem frgp0
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 𝑦 𝑧 𝑛 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgp0.m . . 3 𝐺 = (freeGrp‘𝐼)
2 eqid 2760 . . 3 (freeMnd‘(𝐼 × 2𝑜)) = (freeMnd‘(𝐼 × 2𝑜))
3 frgp0.r . . 3 = ( ~FG𝐼)
41, 2, 3frgpval 18371 . 2 (𝐼𝑉𝐺 = ((freeMnd‘(𝐼 × 2𝑜)) /s ))
5 2on 7737 . . . . 5 2𝑜 ∈ On
6 xpexg 7125 . . . . 5 ((𝐼𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
75, 6mpan2 709 . . . 4 (𝐼𝑉 → (𝐼 × 2𝑜) ∈ V)
8 eqid 2760 . . . . 5 (Base‘(freeMnd‘(𝐼 × 2𝑜))) = (Base‘(freeMnd‘(𝐼 × 2𝑜)))
92, 8frmdbas 17590 . . . 4 ((𝐼 × 2𝑜) ∈ V → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
107, 9syl 17 . . 3 (𝐼𝑉 → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
1110eqcomd 2766 . 2 (𝐼𝑉 → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
12 eqidd 2761 . 2 (𝐼𝑉 → (+g‘(freeMnd‘(𝐼 × 2𝑜))) = (+g‘(freeMnd‘(𝐼 × 2𝑜))))
13 eqid 2760 . . . 4 ( I ‘Word (𝐼 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜))
1413, 3efger 18331 . . 3 Er ( I ‘Word (𝐼 × 2𝑜))
15 wrdexg 13501 . . . . 5 ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
16 fvi 6417 . . . . 5 (Word (𝐼 × 2𝑜) ∈ V → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜))
177, 15, 163syl 18 . . . 4 (𝐼𝑉 → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜))
18 ereq2 7919 . . . 4 (( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜) → ( Er ( I ‘Word (𝐼 × 2𝑜)) ↔ Er Word (𝐼 × 2𝑜)))
1917, 18syl 17 . . 3 (𝐼𝑉 → ( Er ( I ‘Word (𝐼 × 2𝑜)) ↔ Er Word (𝐼 × 2𝑜)))
2014, 19mpbii 223 . 2 (𝐼𝑉 Er Word (𝐼 × 2𝑜))
21 fvexd 6364 . 2 (𝐼𝑉 → (freeMnd‘(𝐼 × 2𝑜)) ∈ V)
22 eqid 2760 . . . 4 (+g‘(freeMnd‘(𝐼 × 2𝑜))) = (+g‘(freeMnd‘(𝐼 × 2𝑜)))
231, 2, 3, 22frgpcpbl 18372 . . 3 ((𝑎 𝑏𝑐 𝑑) → (𝑎(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑐) (𝑏(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑑))
2423a1i 11 . 2 (𝐼𝑉 → ((𝑎 𝑏𝑐 𝑑) → (𝑎(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑐) (𝑏(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑑)))
252frmdmnd 17597 . . . . . 6 ((𝐼 × 2𝑜) ∈ V → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
267, 25syl 17 . . . . 5 (𝐼𝑉 → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
27263ad2ant1 1128 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
28 simp2 1132 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
29113ad2ant1 1128 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
3028, 29eleqtrd 2841 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
31 simp3 1133 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑦 ∈ Word (𝐼 × 2𝑜))
3231, 29eleqtrd 2841 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
338, 22mndcl 17502 . . . 4 (((freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ∧ 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))) → (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
3427, 30, 32, 33syl3anc 1477 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
3534, 29eleqtrrd 2842 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ Word (𝐼 × 2𝑜))
3620adantr 472 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → Er Word (𝐼 × 2𝑜))
3726adantr 472 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
38343adant3r3 1200 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
39 simpr3 1238 . . . . . . 7 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑧 ∈ Word (𝐼 × 2𝑜))
4011adantr 472 . . . . . . 7 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
4139, 40eleqtrd 2841 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
428, 22mndcl 17502 . . . . . 6 (((freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd ∧ (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ∧ 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦)(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
4337, 38, 41, 42syl3anc 1477 . . . . 5 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦)(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
4443, 40eleqtrrd 2842 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦)(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧) ∈ Word (𝐼 × 2𝑜))
4536, 44erref 7931 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦)(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧) ((𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦)(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧))
46303adant3r3 1200 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
47323adant3r3 1200 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
488, 22mndass 17503 . . . 4 (((freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd ∧ (𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ∧ 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ∧ 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦)(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧) = (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))(𝑦(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧)))
4937, 46, 47, 41, 48syl13anc 1479 . . 3 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦)(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧) = (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))(𝑦(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧)))
5045, 49breqtrd 4830 . 2 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → ((𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦)(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧) (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))(𝑦(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑧)))
51 wrd0 13516 . . 3 ∅ ∈ Word (𝐼 × 2𝑜)
5251a1i 11 . 2 (𝐼𝑉 → ∅ ∈ Word (𝐼 × 2𝑜))
5351, 11syl5eleq 2845 . . . . . 6 (𝐼𝑉 → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
5453adantr 472 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
5511eleq2d 2825 . . . . . 6 (𝐼𝑉 → (𝑥 ∈ Word (𝐼 × 2𝑜) ↔ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))))
5655biimpa 502 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
572, 8, 22frmdadd 17593 . . . . 5 ((∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (∅ ++ 𝑥))
5854, 56, 57syl2anc 696 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (∅ ++ 𝑥))
59 ccatlid 13558 . . . . 5 (𝑥 ∈ Word (𝐼 × 2𝑜) → (∅ ++ 𝑥) = 𝑥)
6059adantl 473 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅ ++ 𝑥) = 𝑥)
6158, 60eqtrd 2794 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = 𝑥)
6220adantr 472 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → Er Word (𝐼 × 2𝑜))
63 simpr 479 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
6462, 63erref 7931 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 𝑥)
6561, 64eqbrtrd 4826 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) 𝑥)
66 revcl 13710 . . . 4 (𝑥 ∈ Word (𝐼 × 2𝑜) → (reverse‘𝑥) ∈ Word (𝐼 × 2𝑜))
6766adantl 473 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (reverse‘𝑥) ∈ Word (𝐼 × 2𝑜))
68 eqid 2760 . . . . 5 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
6968efgmf 18326 . . . 4 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩):(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
7069a1i 11 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩):(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜))
71 wrdco 13777 . . 3 (((reverse‘𝑥) ∈ Word (𝐼 × 2𝑜) ∧ (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩):(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)) → ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2𝑜))
7267, 70, 71syl2anc 696 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2𝑜))
7311adantr 472 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
7472, 73eleqtrd 2841 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
752, 8, 22frmdadd 17593 . . . 4 ((((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))) → (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥))(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥))
7674, 56, 75syl2anc 696 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥))(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥))
7717eleq2d 2825 . . . . 5 (𝐼𝑉 → (𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↔ 𝑥 ∈ Word (𝐼 × 2𝑜)))
7877biimpar 503 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)))
79 eqid 2760 . . . . 5 (𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩))) = (𝑣 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘𝑤)”⟩⟩)))
8013, 3, 68, 79efginvrel1 18341 . . . 4 (𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)) → (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∅)
8178, 80syl 17 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∅)
8276, 81eqbrtrd 4826 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥))(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) ∅)
834, 11, 12, 20, 21, 24, 35, 50, 52, 65, 72, 82qusgrp2 17734 1 (𝐼𝑉 → (𝐺 ∈ Grp ∧ [∅] = (0g𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  Vcvv 3340   ∖ cdif 3712  ∅c0 4058  ⟨cop 4327  ⟨cotp 4329   class class class wbr 4804   ↦ cmpt 4881   I cid 5173   × cxp 5264   ∘ ccom 5270  Oncon0 5884  ⟶wf 6045  ‘cfv 6049  (class class class)co 6813   ↦ cmpt2 6815  1𝑜c1o 7722  2𝑜c2o 7723   Er wer 7908  [cec 7909  0cc0 10128  ...cfz 12519  ♯chash 13311  Word cword 13477   ++ cconcat 13479   splice csplice 13482  reversecreverse 13483  ⟨“cs2 13786  Basecbs 16059  +gcplusg 16143  0gc0g 16302  Mndcmnd 17495  freeMndcfrmd 17585  Grpcgrp 17623   ~FG cefg 18319  freeGrpcfrgp 18320 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-ec 7913  df-qs 7917  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-xnn0 11556  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-fzo 12660  df-hash 13312  df-word 13485  df-lsw 13486  df-concat 13487  df-s1 13488  df-substr 13489  df-splice 13490  df-reverse 13491  df-s2 13793  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-0g 16304  df-imas 16370  df-qus 16371  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-frmd 17587  df-grp 17626  df-efg 18322  df-frgp 18323 This theorem is referenced by:  frgpgrp  18375  frgpinv  18377  frgpmhm  18378
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