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Theorem frgpup1 18011
 Description: Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Hypotheses
Ref Expression
frgpup.b 𝐵 = (Base‘𝐻)
frgpup.n 𝑁 = (invg𝐻)
frgpup.t 𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
frgpup.h (𝜑𝐻 ∈ Grp)
frgpup.i (𝜑𝐼𝑉)
frgpup.a (𝜑𝐹:𝐼𝐵)
frgpup.w 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
frgpup.r = ( ~FG𝐼)
frgpup.g 𝐺 = (freeGrp‘𝐼)
frgpup.x 𝑋 = (Base‘𝐺)
frgpup.e 𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)
Assertion
Ref Expression
frgpup1 (𝜑𝐸 ∈ (𝐺 GrpHom 𝐻))
Distinct variable groups:   𝑦,𝑔,𝑧   𝑔,𝐻   𝑦,𝐹,𝑧   𝑦,𝑁,𝑧   𝐵,𝑔,𝑦,𝑧   𝑇,𝑔   ,𝑔   𝜑,𝑔,𝑦,𝑧   𝑦,𝐼,𝑧   𝑔,𝑊
Allowed substitution hints:   (𝑦,𝑧)   𝑇(𝑦,𝑧)   𝐸(𝑦,𝑧,𝑔)   𝐹(𝑔)   𝐺(𝑦,𝑧,𝑔)   𝐻(𝑦,𝑧)   𝐼(𝑔)   𝑁(𝑔)   𝑉(𝑦,𝑧,𝑔)   𝑊(𝑦,𝑧)   𝑋(𝑦,𝑧,𝑔)

Proof of Theorem frgpup1
Dummy variables 𝑎 𝑢 𝑐 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpup.x . 2 𝑋 = (Base‘𝐺)
2 frgpup.b . 2 𝐵 = (Base‘𝐻)
3 eqid 2610 . 2 (+g𝐺) = (+g𝐺)
4 eqid 2610 . 2 (+g𝐻) = (+g𝐻)
5 frgpup.i . . 3 (𝜑𝐼𝑉)
6 frgpup.g . . . 4 𝐺 = (freeGrp‘𝐼)
76frgpgrp 17998 . . 3 (𝐼𝑉𝐺 ∈ Grp)
85, 7syl 17 . 2 (𝜑𝐺 ∈ Grp)
9 frgpup.h . 2 (𝜑𝐻 ∈ Grp)
10 frgpup.n . . 3 𝑁 = (invg𝐻)
11 frgpup.t . . 3 𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
12 frgpup.a . . 3 (𝜑𝐹:𝐼𝐵)
13 frgpup.w . . 3 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
14 frgpup.r . . 3 = ( ~FG𝐼)
15 frgpup.e . . 3 𝐸 = ran (𝑔𝑊 ↦ ⟨[𝑔] , (𝐻 Σg (𝑇𝑔))⟩)
162, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupf 18009 . 2 (𝜑𝐸:𝑋𝐵)
17 eqid 2610 . . . . . . . . . . 11 (freeMnd‘(𝐼 × 2𝑜)) = (freeMnd‘(𝐼 × 2𝑜))
186, 17, 14frgpval 17994 . . . . . . . . . 10 (𝐼𝑉𝐺 = ((freeMnd‘(𝐼 × 2𝑜)) /s ))
195, 18syl 17 . . . . . . . . 9 (𝜑𝐺 = ((freeMnd‘(𝐼 × 2𝑜)) /s ))
20 2on 7455 . . . . . . . . . . . . 13 2𝑜 ∈ On
21 xpexg 6858 . . . . . . . . . . . . 13 ((𝐼𝑉 ∧ 2𝑜 ∈ On) → (𝐼 × 2𝑜) ∈ V)
225, 20, 21sylancl 693 . . . . . . . . . . . 12 (𝜑 → (𝐼 × 2𝑜) ∈ V)
23 wrdexg 13170 . . . . . . . . . . . 12 ((𝐼 × 2𝑜) ∈ V → Word (𝐼 × 2𝑜) ∈ V)
24 fvi 6165 . . . . . . . . . . . 12 (Word (𝐼 × 2𝑜) ∈ V → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜))
2522, 23, 243syl 18 . . . . . . . . . . 11 (𝜑 → ( I ‘Word (𝐼 × 2𝑜)) = Word (𝐼 × 2𝑜))
2613, 25syl5eq 2656 . . . . . . . . . 10 (𝜑𝑊 = Word (𝐼 × 2𝑜))
27 eqid 2610 . . . . . . . . . . . 12 (Base‘(freeMnd‘(𝐼 × 2𝑜))) = (Base‘(freeMnd‘(𝐼 × 2𝑜)))
2817, 27frmdbas 17212 . . . . . . . . . . 11 ((𝐼 × 2𝑜) ∈ V → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
2922, 28syl 17 . . . . . . . . . 10 (𝜑 → (Base‘(freeMnd‘(𝐼 × 2𝑜))) = Word (𝐼 × 2𝑜))
3026, 29eqtr4d 2647 . . . . . . . . 9 (𝜑𝑊 = (Base‘(freeMnd‘(𝐼 × 2𝑜))))
31 fvex 6113 . . . . . . . . . . 11 ( ~FG𝐼) ∈ V
3214, 31eqeltri 2684 . . . . . . . . . 10 ∈ V
3332a1i 11 . . . . . . . . 9 (𝜑 ∈ V)
34 fvex 6113 . . . . . . . . . 10 (freeMnd‘(𝐼 × 2𝑜)) ∈ V
3534a1i 11 . . . . . . . . 9 (𝜑 → (freeMnd‘(𝐼 × 2𝑜)) ∈ V)
3619, 30, 33, 35qusbas 16028 . . . . . . . 8 (𝜑 → (𝑊 / ) = (Base‘𝐺))
3736, 1syl6reqr 2663 . . . . . . 7 (𝜑𝑋 = (𝑊 / ))
38 eqimss 3620 . . . . . . 7 (𝑋 = (𝑊 / ) → 𝑋 ⊆ (𝑊 / ))
3937, 38syl 17 . . . . . 6 (𝜑𝑋 ⊆ (𝑊 / ))
4039adantr 480 . . . . 5 ((𝜑𝑎𝑋) → 𝑋 ⊆ (𝑊 / ))
4140sselda 3568 . . . 4 (((𝜑𝑎𝑋) ∧ 𝑐𝑋) → 𝑐 ∈ (𝑊 / ))
42 eqid 2610 . . . . 5 (𝑊 / ) = (𝑊 / )
43 oveq2 6557 . . . . . . 7 ([𝑢] = 𝑐 → (𝑎(+g𝐺)[𝑢] ) = (𝑎(+g𝐺)𝑐))
4443fveq2d 6107 . . . . . 6 ([𝑢] = 𝑐 → (𝐸‘(𝑎(+g𝐺)[𝑢] )) = (𝐸‘(𝑎(+g𝐺)𝑐)))
45 fveq2 6103 . . . . . . 7 ([𝑢] = 𝑐 → (𝐸‘[𝑢] ) = (𝐸𝑐))
4645oveq2d 6565 . . . . . 6 ([𝑢] = 𝑐 → ((𝐸𝑎)(+g𝐻)(𝐸‘[𝑢] )) = ((𝐸𝑎)(+g𝐻)(𝐸𝑐)))
4744, 46eqeq12d 2625 . . . . 5 ([𝑢] = 𝑐 → ((𝐸‘(𝑎(+g𝐺)[𝑢] )) = ((𝐸𝑎)(+g𝐻)(𝐸‘[𝑢] )) ↔ (𝐸‘(𝑎(+g𝐺)𝑐)) = ((𝐸𝑎)(+g𝐻)(𝐸𝑐))))
4839sselda 3568 . . . . . . . 8 ((𝜑𝑎𝑋) → 𝑎 ∈ (𝑊 / ))
4948adantlr 747 . . . . . . 7 (((𝜑𝑢𝑊) ∧ 𝑎𝑋) → 𝑎 ∈ (𝑊 / ))
50 oveq1 6556 . . . . . . . . . 10 ([𝑡] = 𝑎 → ([𝑡] (+g𝐺)[𝑢] ) = (𝑎(+g𝐺)[𝑢] ))
5150fveq2d 6107 . . . . . . . . 9 ([𝑡] = 𝑎 → (𝐸‘([𝑡] (+g𝐺)[𝑢] )) = (𝐸‘(𝑎(+g𝐺)[𝑢] )))
52 fveq2 6103 . . . . . . . . . 10 ([𝑡] = 𝑎 → (𝐸‘[𝑡] ) = (𝐸𝑎))
5352oveq1d 6564 . . . . . . . . 9 ([𝑡] = 𝑎 → ((𝐸‘[𝑡] )(+g𝐻)(𝐸‘[𝑢] )) = ((𝐸𝑎)(+g𝐻)(𝐸‘[𝑢] )))
5451, 53eqeq12d 2625 . . . . . . . 8 ([𝑡] = 𝑎 → ((𝐸‘([𝑡] (+g𝐺)[𝑢] )) = ((𝐸‘[𝑡] )(+g𝐻)(𝐸‘[𝑢] )) ↔ (𝐸‘(𝑎(+g𝐺)[𝑢] )) = ((𝐸𝑎)(+g𝐻)(𝐸‘[𝑢] ))))
55 fviss 6166 . . . . . . . . . . . . . . . 16 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
5613, 55eqsstri 3598 . . . . . . . . . . . . . . 15 𝑊 ⊆ Word (𝐼 × 2𝑜)
5756sseli 3564 . . . . . . . . . . . . . 14 (𝑡𝑊𝑡 ∈ Word (𝐼 × 2𝑜))
5856sseli 3564 . . . . . . . . . . . . . 14 (𝑢𝑊𝑢 ∈ Word (𝐼 × 2𝑜))
59 ccatcl 13212 . . . . . . . . . . . . . 14 ((𝑡 ∈ Word (𝐼 × 2𝑜) ∧ 𝑢 ∈ Word (𝐼 × 2𝑜)) → (𝑡 ++ 𝑢) ∈ Word (𝐼 × 2𝑜))
6057, 58, 59syl2an 493 . . . . . . . . . . . . 13 ((𝑡𝑊𝑢𝑊) → (𝑡 ++ 𝑢) ∈ Word (𝐼 × 2𝑜))
6113efgrcl 17951 . . . . . . . . . . . . . . 15 (𝑡𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
6261adantr 480 . . . . . . . . . . . . . 14 ((𝑡𝑊𝑢𝑊) → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
6362simprd 478 . . . . . . . . . . . . 13 ((𝑡𝑊𝑢𝑊) → 𝑊 = Word (𝐼 × 2𝑜))
6460, 63eleqtrrd 2691 . . . . . . . . . . . 12 ((𝑡𝑊𝑢𝑊) → (𝑡 ++ 𝑢) ∈ 𝑊)
652, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupval 18010 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 ++ 𝑢) ∈ 𝑊) → (𝐸‘[(𝑡 ++ 𝑢)] ) = (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))))
6664, 65sylan2 490 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝐸‘[(𝑡 ++ 𝑢)] ) = (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))))
6757ad2antrl 760 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → 𝑡 ∈ Word (𝐼 × 2𝑜))
6858ad2antll 761 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → 𝑢 ∈ Word (𝐼 × 2𝑜))
692, 10, 11, 9, 5, 12frgpuptf 18006 . . . . . . . . . . . . . 14 (𝜑𝑇:(𝐼 × 2𝑜)⟶𝐵)
7069adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → 𝑇:(𝐼 × 2𝑜)⟶𝐵)
71 ccatco 13432 . . . . . . . . . . . . 13 ((𝑡 ∈ Word (𝐼 × 2𝑜) ∧ 𝑢 ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇 ∘ (𝑡 ++ 𝑢)) = ((𝑇𝑡) ++ (𝑇𝑢)))
7267, 68, 70, 71syl3anc 1318 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝑇 ∘ (𝑡 ++ 𝑢)) = ((𝑇𝑡) ++ (𝑇𝑢)))
7372oveq2d 6565 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝐻 Σg (𝑇 ∘ (𝑡 ++ 𝑢))) = (𝐻 Σg ((𝑇𝑡) ++ (𝑇𝑢))))
74 grpmnd 17252 . . . . . . . . . . . . . 14 (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
759, 74syl 17 . . . . . . . . . . . . 13 (𝜑𝐻 ∈ Mnd)
7675adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → 𝐻 ∈ Mnd)
77 wrdco 13428 . . . . . . . . . . . . . 14 ((𝑡 ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇𝑡) ∈ Word 𝐵)
7857, 69, 77syl2anr 494 . . . . . . . . . . . . 13 ((𝜑𝑡𝑊) → (𝑇𝑡) ∈ Word 𝐵)
7978adantrr 749 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝑇𝑡) ∈ Word 𝐵)
80 wrdco 13428 . . . . . . . . . . . . 13 ((𝑢 ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇𝑢) ∈ Word 𝐵)
8168, 70, 80syl2anc 691 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝑇𝑢) ∈ Word 𝐵)
822, 4gsumccat 17201 . . . . . . . . . . . 12 ((𝐻 ∈ Mnd ∧ (𝑇𝑡) ∈ Word 𝐵 ∧ (𝑇𝑢) ∈ Word 𝐵) → (𝐻 Σg ((𝑇𝑡) ++ (𝑇𝑢))) = ((𝐻 Σg (𝑇𝑡))(+g𝐻)(𝐻 Σg (𝑇𝑢))))
8376, 79, 81, 82syl3anc 1318 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝐻 Σg ((𝑇𝑡) ++ (𝑇𝑢))) = ((𝐻 Σg (𝑇𝑡))(+g𝐻)(𝐻 Σg (𝑇𝑢))))
8466, 73, 833eqtrd 2648 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝐸‘[(𝑡 ++ 𝑢)] ) = ((𝐻 Σg (𝑇𝑡))(+g𝐻)(𝐻 Σg (𝑇𝑢))))
8513, 6, 14, 3frgpadd 17999 . . . . . . . . . . . 12 ((𝑡𝑊𝑢𝑊) → ([𝑡] (+g𝐺)[𝑢] ) = [(𝑡 ++ 𝑢)] )
8685adantl 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → ([𝑡] (+g𝐺)[𝑢] ) = [(𝑡 ++ 𝑢)] )
8786fveq2d 6107 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝐸‘([𝑡] (+g𝐺)[𝑢] )) = (𝐸‘[(𝑡 ++ 𝑢)] ))
882, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupval 18010 . . . . . . . . . . . 12 ((𝜑𝑡𝑊) → (𝐸‘[𝑡] ) = (𝐻 Σg (𝑇𝑡)))
8988adantrr 749 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝐸‘[𝑡] ) = (𝐻 Σg (𝑇𝑡)))
902, 10, 11, 9, 5, 12, 13, 14, 6, 1, 15frgpupval 18010 . . . . . . . . . . . 12 ((𝜑𝑢𝑊) → (𝐸‘[𝑢] ) = (𝐻 Σg (𝑇𝑢)))
9190adantrl 748 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝐸‘[𝑢] ) = (𝐻 Σg (𝑇𝑢)))
9289, 91oveq12d 6567 . . . . . . . . . 10 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → ((𝐸‘[𝑡] )(+g𝐻)(𝐸‘[𝑢] )) = ((𝐻 Σg (𝑇𝑡))(+g𝐻)(𝐻 Σg (𝑇𝑢))))
9384, 87, 923eqtr4d 2654 . . . . . . . . 9 ((𝜑 ∧ (𝑡𝑊𝑢𝑊)) → (𝐸‘([𝑡] (+g𝐺)[𝑢] )) = ((𝐸‘[𝑡] )(+g𝐻)(𝐸‘[𝑢] )))
9493anass1rs 845 . . . . . . . 8 (((𝜑𝑢𝑊) ∧ 𝑡𝑊) → (𝐸‘([𝑡] (+g𝐺)[𝑢] )) = ((𝐸‘[𝑡] )(+g𝐻)(𝐸‘[𝑢] )))
9542, 54, 94ectocld 7701 . . . . . . 7 (((𝜑𝑢𝑊) ∧ 𝑎 ∈ (𝑊 / )) → (𝐸‘(𝑎(+g𝐺)[𝑢] )) = ((𝐸𝑎)(+g𝐻)(𝐸‘[𝑢] )))
9649, 95syldan 486 . . . . . 6 (((𝜑𝑢𝑊) ∧ 𝑎𝑋) → (𝐸‘(𝑎(+g𝐺)[𝑢] )) = ((𝐸𝑎)(+g𝐻)(𝐸‘[𝑢] )))
9796an32s 842 . . . . 5 (((𝜑𝑎𝑋) ∧ 𝑢𝑊) → (𝐸‘(𝑎(+g𝐺)[𝑢] )) = ((𝐸𝑎)(+g𝐻)(𝐸‘[𝑢] )))
9842, 47, 97ectocld 7701 . . . 4 (((𝜑𝑎𝑋) ∧ 𝑐 ∈ (𝑊 / )) → (𝐸‘(𝑎(+g𝐺)𝑐)) = ((𝐸𝑎)(+g𝐻)(𝐸𝑐)))
9941, 98syldan 486 . . 3 (((𝜑𝑎𝑋) ∧ 𝑐𝑋) → (𝐸‘(𝑎(+g𝐺)𝑐)) = ((𝐸𝑎)(+g𝐻)(𝐸𝑐)))
10099anasss 677 . 2 ((𝜑 ∧ (𝑎𝑋𝑐𝑋)) → (𝐸‘(𝑎(+g𝐺)𝑐)) = ((𝐸𝑎)(+g𝐻)(𝐸𝑐)))
1011, 2, 3, 4, 8, 9, 16, 100isghmd 17492 1 (𝜑𝐸 ∈ (𝐺 GrpHom 𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ifcif 4036  ⟨cop 4131   ↦ cmpt 4643   I cid 4948   × cxp 5036  ran crn 5039   ∘ ccom 5042  Oncon0 5640  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  2𝑜c2o 7441  [cec 7627   / cqs 7628  Word cword 13146   ++ cconcat 13148  Basecbs 15695  +gcplusg 15768   Σg cgsu 15924   /s cqus 15988  Mndcmnd 17117  freeMndcfrmd 17207  Grpcgrp 17245  invgcminusg 17246   GrpHom cghm 17480   ~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-seq 12664  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-sets 15701  df-ress 15702  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-gsum 15926  df-imas 15991  df-qus 15992  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-frmd 17209  df-grp 17248  df-minusg 17249  df-ghm 17481  df-efg 17945  df-frgp 17946 This theorem is referenced by:  frgpup3lem  18013  frgpup3  18014
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