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Theorem frgp0 18380
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 2771 . . 3 (freeMnd‘(𝐼 × 2𝑜)) = (freeMnd‘(𝐼 × 2𝑜))
3 frgp0.r . . 3 = ( ~FG𝐼)
41, 2, 3frgpval 18378 . 2 (𝐼𝑉𝐺 = ((freeMnd‘(𝐼 × 2𝑜)) /s ))
5 2on 7726 . . . . 5 2𝑜 ∈ On
6 xpexg 7111 . . . . 5 ((𝐼𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
75, 6mpan2 671 . . . 4 (𝐼𝑉 → (𝐼 × 2𝑜) ∈ V)
8 eqid 2771 . . . . 5 (Base‘(freeMnd‘(𝐼 × 2𝑜))) = (Base‘(freeMnd‘(𝐼 × 2𝑜)))
92, 8frmdbas 17597 . . . 4 ((𝐼 × 2𝑜) ∈ V → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
107, 9syl 17 . . 3 (𝐼𝑉 → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
1110eqcomd 2777 . 2 (𝐼𝑉 → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
12 eqidd 2772 . 2 (𝐼𝑉 → (+g‘(freeMnd‘(𝐼 × 2𝑜))) = (+g‘(freeMnd‘(𝐼 × 2𝑜))))
13 eqid 2771 . . . 4 ( I ‘Word (𝐼 × 2𝑜)) = ( I ‘Word (𝐼 × 2𝑜))
1413, 3efger 18338 . . 3 Er ( I ‘Word (𝐼 × 2𝑜))
15 wrdexg 13511 . . . . 5 ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
16 fvi 6399 . . . . 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 7908 . . . 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 6346 . 2 (𝐼𝑉 → (freeMnd‘(𝐼 × 2𝑜)) ∈ V)
22 eqid 2771 . . . 4 (+g‘(freeMnd‘(𝐼 × 2𝑜))) = (+g‘(freeMnd‘(𝐼 × 2𝑜)))
231, 2, 3, 22frgpcpbl 18379 . . 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 17604 . . . . . 6 ((𝐼 × 2𝑜) ∈ V → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
267, 25syl 17 . . . . 5 (𝐼𝑉 → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
27263ad2ant1 1127 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
28 simp2 1131 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
29113ad2ant1 1127 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
3028, 29eleqtrd 2852 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
31 simp3 1132 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑦 ∈ Word (𝐼 × 2𝑜))
3231, 29eleqtrd 2852 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
338, 22mndcl 17509 . . . 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 1476 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
3534, 29eleqtrrd 2853 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜)) → (𝑥(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑦) ∈ Word (𝐼 × 2𝑜))
3620adantr 466 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → Er Word (𝐼 × 2𝑜))
3726adantr 466 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → (freeMnd‘(𝐼 × 2𝑜)) ∈ Mnd)
38343adant3r3 1199 . . . . . 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 1237 . . . . . . 7 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑧 ∈ Word (𝐼 × 2𝑜))
4011adantr 466 . . . . . . 7 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
4139, 40eleqtrd 2852 . . . . . 6 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑧 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
428, 22mndcl 17509 . . . . . 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 1476 . . . . 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 2853 . . . 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 7920 . . 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 1199 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
47323adant3r3 1199 . . . 4 ((𝐼𝑉 ∧ (𝑥 ∈ Word (𝐼 × 2𝑜) ∧ 𝑦 ∈ Word (𝐼 × 2𝑜) ∧ 𝑧 ∈ Word (𝐼 × 2𝑜))) → 𝑦 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
488, 22mndass 17510 . . . 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 1478 . . 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 4813 . 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 13526 . . 3 ∅ ∈ Word (𝐼 × 2𝑜)
5251a1i 11 . 2 (𝐼𝑉 → ∅ ∈ Word (𝐼 × 2𝑜))
5351, 11syl5eleq 2856 . . . . . 6 (𝐼𝑉 → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
5453adantr 466 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → ∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
5511eleq2d 2836 . . . . . 6 (𝐼𝑉 → (𝑥 ∈ Word (𝐼 × 2𝑜) ↔ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))))
5655biimpa 462 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
572, 8, 22frmdadd 17600 . . . . 5 ((∅ ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))) ∧ 𝑥 ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜)))) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (∅ ++ 𝑥))
5854, 56, 57syl2anc 573 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = (∅ ++ 𝑥))
59 ccatlid 13568 . . . . 5 (𝑥 ∈ Word (𝐼 × 2𝑜) → (∅ ++ 𝑥) = 𝑥)
6059adantl 467 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅ ++ 𝑥) = 𝑥)
6158, 60eqtrd 2805 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) = 𝑥)
6220adantr 466 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → Er Word (𝐼 × 2𝑜))
63 simpr 471 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
6462, 63erref 7920 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 𝑥)
6561, 64eqbrtrd 4809 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (∅(+g‘(freeMnd‘(𝐼 × 2𝑜)))𝑥) 𝑥)
66 revcl 13719 . . . 4 (𝑥 ∈ Word (𝐼 × 2𝑜) → (reverse‘𝑥) ∈ Word (𝐼 × 2𝑜))
6766adantl 467 . . 3 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → (reverse‘𝑥) ∈ Word (𝐼 × 2𝑜))
68 eqid 2771 . . . . 5 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
6968efgmf 18333 . . . 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 13786 . . 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 573 . 2 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ∈ Word (𝐼 × 2𝑜))
7311adantr 466 . . . . 5 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → Word (𝐼 × 2𝑜) = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
7472, 73eleqtrd 2852 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) ∘ (reverse‘𝑥)) ∈ (Base‘(freeMnd‘(𝐼 × 2𝑜))))
752, 8, 22frmdadd 17600 . . . 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 573 . . 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 2836 . . . . 5 (𝐼𝑉 → (𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)) ↔ 𝑥 ∈ Word (𝐼 × 2𝑜)))
7877biimpar 463 . . . 4 ((𝐼𝑉𝑥 ∈ Word (𝐼 × 2𝑜)) → 𝑥 ∈ ( I ‘Word (𝐼 × 2𝑜)))
79 eqid 2771 . . . . 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 18348 . . . 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 4809 . 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 17741 1 (𝐼𝑉 → (𝐺 ∈ Grp ∧ [∅] = (0g𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  cdif 3720  c0 4063  cop 4323  cotp 4325   class class class wbr 4787  cmpt 4864   I cid 5157   × cxp 5248  ccom 5254  Oncon0 5865  wf 6026  cfv 6030  (class class class)co 6796  cmpt2 6798  1𝑜c1o 7710  2𝑜c2o 7711   Er wer 7897  [cec 7898  0cc0 10142  ...cfz 12533  chash 13321  Word cword 13487   ++ cconcat 13489   splice csplice 13492  reversecreverse 13493  ⟨“cs2 13795  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502  freeMndcfrmd 17592  Grpcgrp 17630   ~FG cefg 18326  freeGrpcfrgp 18327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-cnex 10198  ax-resscn 10199  ax-1cn 10200  ax-icn 10201  ax-addcl 10202  ax-addrcl 10203  ax-mulcl 10204  ax-mulrcl 10205  ax-mulcom 10206  ax-addass 10207  ax-mulass 10208  ax-distr 10209  ax-i2m1 10210  ax-1ne0 10211  ax-1rid 10212  ax-rnegex 10213  ax-rrecex 10214  ax-cnre 10215  ax-pre-lttri 10216  ax-pre-lttrn 10217  ax-pre-ltadd 10218  ax-pre-mulgt0 10219
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-ot 4326  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-er 7900  df-ec 7902  df-qs 7906  df-map 8015  df-pm 8016  df-en 8114  df-dom 8115  df-sdom 8116  df-fin 8117  df-sup 8508  df-inf 8509  df-card 8969  df-pnf 10282  df-mnf 10283  df-xr 10284  df-ltxr 10285  df-le 10286  df-sub 10474  df-neg 10475  df-nn 11227  df-2 11285  df-3 11286  df-4 11287  df-5 11288  df-6 11289  df-7 11290  df-8 11291  df-9 11292  df-n0 11500  df-xnn0 11571  df-z 11585  df-dec 11701  df-uz 11894  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-lsw 13496  df-concat 13497  df-s1 13498  df-substr 13499  df-splice 13500  df-reverse 13501  df-s2 13802  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-plusg 16162  df-mulr 16163  df-sca 16165  df-vsca 16166  df-ip 16167  df-tset 16168  df-ple 16169  df-ds 16172  df-0g 16310  df-imas 16376  df-qus 16377  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-frmd 17594  df-grp 17633  df-efg 18329  df-frgp 18330
This theorem is referenced by:  frgpgrp  18382  frgpinv  18384  frgpmhm  18385
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