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Theorem frgp0 17996
 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 2610 . . 3 (freeMnd‘(𝐼 × 2𝑜)) = (freeMnd‘(𝐼 × 2𝑜))
3 frgp0.r . . 3 = ( ~FG𝐼)
41, 2, 3frgpval 17994 . 2 (𝐼𝑉𝐺 = ((freeMnd‘(𝐼 × 2𝑜)) /s ))
5 2on 7455 . . . . 5 2𝑜 ∈ On
6 xpexg 6858 . . . . 5 ((𝐼𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
75, 6mpan2 703 . . . 4 (𝐼𝑉 → (𝐼 × 2𝑜) ∈ V)
8 eqid 2610 . . . . 5 (Base‘(freeMnd‘(𝐼 × 2𝑜))) = (Base‘(freeMnd‘(𝐼 × 2𝑜)))
92, 8frmdbas 17212 . . . 4 ((𝐼 × 2𝑜) ∈ V → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
107, 9syl 17 . . 3 (𝐼𝑉 → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
1110eqcomd 2616 . 2 (𝐼𝑉 → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
12 eqidd 2611 . 2 (𝐼𝑉 → (+g‘(freeMnd‘(𝐼 × 2𝑜))) = (+g‘(freeMnd‘(𝐼 × 2𝑜))))
13 eqid 2610 . . . 4 ( I ‘Word (𝐼 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜))
1413, 3efger 17954 . . 3 Er ( I ‘Word (𝐼 × 2𝑜))
15 wrdexg 13170 . . . . 5 ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
16 fvi 6165 . . . . 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 7637 . . . 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 222 . 2 (𝐼𝑉 Er Word (𝐼 × 2𝑜))
21 fvex 6113 . . 3 (freeMnd‘(𝐼 × 2𝑜)) ∈ V
2221a1i 11 . 2 (𝐼𝑉 → (freeMnd‘(𝐼 × 2𝑜)) ∈ V)
23 eqid 2610 . . . 4 (+g‘(freeMnd‘(𝐼 × 2𝑜))) = (+g‘(freeMnd‘(𝐼 × 2𝑜)))
241, 2, 3, 23frgpcpbl 17995 . . 3 ((𝑎 𝑏𝑐 𝑑) → (𝑎(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑐) (𝑏(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑑))
2524a1i 11 . 2 (𝐼𝑉 → ((𝑎 𝑏𝑐 𝑑) → (𝑎(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑐) (𝑏(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑑)))
262frmdmnd 17219 . . . . . 6 ((𝐼 × 2𝑜) ∈ V → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
277, 26syl 17 . . . . 5 (𝐼𝑉 → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
28273ad2ant1 1075 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
29 simp2 1055 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
30113ad2ant1 1075 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
3129, 30eleqtrd 2690 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
32 simp3 1056 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑦 ∈ Word (𝐼 × 2𝑜))
3332, 30eleqtrd 2690 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
348, 23mndcl 17124 . . . 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))))
3528, 31, 33, 34syl3anc 1318 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
3635, 30eleqtrrd 2691 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ Word (𝐼 × 2𝑜))
3720adantr 480 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → Er Word (𝐼 × 2𝑜))
3827adantr 480 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
39353adant3r3 1268 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
40 simpr3 1062 . . . . . . 7 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑧 ∈ Word (𝐼 × 2𝑜))
4111adantr 480 . . . . . . 7 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
4240, 41eleqtrd 2690 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
438, 23mndcl 17124 . . . . . 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))))
4438, 39, 42, 43syl3anc 1318 . . . . 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))))
4544, 41eleqtrrd 2691 . . . 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))
4637, 45erref 7649 . . 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))
47313adant3r3 1268 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
48333adant3r3 1268 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
498, 23mndass 17125 . . . 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)))
5038, 47, 48, 42, 49syl13anc 1320 . . 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)))
5146, 50breqtrd 4609 . 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)))
52 wrd0 13185 . . 3 ∅ ∈ Word (𝐼 × 2𝑜)
5352a1i 11 . 2 (𝐼𝑉 → ∅ ∈ Word (𝐼 × 2𝑜))
5452, 11syl5eleq 2694 . . . . . 6 (𝐼𝑉 → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
5554adantr 480 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
5611eleq2d 2673 . . . . . 6 (𝐼𝑉 → (𝑥 ∈ Word (𝐼 × 2𝑜) ↔ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))))
5756biimpa 500 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
582, 8, 23frmdadd 17215 . . . . 5 ((∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (∅ ++ 𝑥))
5955, 57, 58syl2anc 691 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (∅ ++ 𝑥))
60 ccatlid 13222 . . . . 5 (𝑥 ∈ Word (𝐼 × 2𝑜) → (∅ ++ 𝑥) = 𝑥)
6160adantl 481 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅ ++ 𝑥) = 𝑥)
6259, 61eqtrd 2644 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = 𝑥)
6320adantr 480 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → Er Word (𝐼 × 2𝑜))
64 simpr 476 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
6563, 64erref 7649 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 𝑥)
6662, 65eqbrtrd 4605 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) 𝑥)
67 revcl 13361 . . . 4 (𝑥 ∈ Word (𝐼 × 2𝑜) → (reverse‘𝑥) ∈ Word (𝐼 × 2𝑜))
6867adantl 481 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (reverse‘𝑥) ∈ Word (𝐼 × 2𝑜))
69 eqid 2610 . . . . 5 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
7069efgmf 17949 . . . 4 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩):(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
7170a1i 11 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩):(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜))
72 wrdco 13428 . . 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))
7368, 71, 72syl2anc 691 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2𝑜))
7411adantr 480 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
7573, 74eleqtrd 2690 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
762, 8, 23frmdadd 17215 . . . 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))
7775, 57, 76syl2anc 691 . . 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))
7817eleq2d 2673 . . . . 5 (𝐼𝑉 → (𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↔ 𝑥 ∈ Word (𝐼 × 2𝑜)))
7978biimpar 501 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)))
80 eqid 2610 . . . . 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)”⟩⟩)))
8113, 3, 69, 80efginvrel1 17964 . . . 4 (𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)) → (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∅)
8279, 81syl 17 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ++ 𝑥) ∅)
8377, 82eqbrtrd 4605 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥))(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) ∅)
844, 11, 12, 20, 22, 25, 36, 51, 53, 66, 73, 83qusgrp2 17356 1 (𝐼𝑉 → (𝐺 ∈ Grp ∧ [∅] = (0g𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  ⟨cop 4131  ⟨cotp 4133   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   × cxp 5036   ∘ ccom 5042  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1𝑜c1o 7440  2𝑜c2o 7441   Er wer 7626  [cec 7627  0cc0 9815  ...cfz 12197  #chash 12979  Word cword 13146   ++ cconcat 13148   splice csplice 13151  reversecreverse 13152  ⟨“cs2 13437  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Mndcmnd 17117  freeMndcfrmd 17207  Grpcgrp 17245   ~FG cefg 17942  freeGrpcfrgp 17943 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-xnn0 11241  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-reverse 13160  df-s2 13444  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-0g 15925  df-imas 15991  df-qus 15992  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-frmd 17209  df-grp 17248  df-efg 17945  df-frgp 17946 This theorem is referenced by:  frgpgrp  17998  frgpinv  18000  frgpmhm  18001
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