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Theorem frgpuplem 18970
 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.)
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)
Assertion
Ref Expression
frgpuplem ((𝜑𝐴 𝐶) → (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑦,𝐹,𝑧   𝑦,𝑁,𝑧   𝑦,𝐵,𝑧   𝜑,𝑦,𝑧   𝑦,𝐼,𝑧
Allowed substitution hints:   𝐶(𝑦,𝑧)   (𝑦,𝑧)   𝑇(𝑦,𝑧)   𝐻(𝑦,𝑧)   𝑉(𝑦,𝑧)   𝑊(𝑦,𝑧)

Proof of Theorem frgpuplem
Dummy variables 𝑎 𝑏 𝑢 𝑣 𝑛 𝑟 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 frgpup.w . . . . . . 7 𝑊 = ( I ‘Word (𝐼 × 2o))
2 frgpup.r . . . . . . 7 = ( ~FG𝐼)
31, 2efgval 18915 . . . . . 6 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))}
4 coeq2 5703 . . . . . . . . . . . . 13 (𝑢 = 𝑣 → (𝑇𝑢) = (𝑇𝑣))
54oveq2d 7171 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))
6 eqid 2758 . . . . . . . . . . . 12 {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} = {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))}
75, 6eqer 8339 . . . . . . . . . . 11 {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} Er V
87a1i 11 . . . . . . . . . 10 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} Er V)
9 ssv 3918 . . . . . . . . . . 11 𝑊 ⊆ V
109a1i 11 . . . . . . . . . 10 (𝜑𝑊 ⊆ V)
118, 10erinxp 8386 . . . . . . . . 9 (𝜑 → ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊)) Er 𝑊)
12 df-xp 5533 . . . . . . . . . . . . 13 (𝑊 × 𝑊) = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑊𝑣𝑊)}
1312ineq1i 4115 . . . . . . . . . . . 12 ((𝑊 × 𝑊) ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))}) = ({⟨𝑢, 𝑣⟩ ∣ (𝑢𝑊𝑣𝑊)} ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))})
14 incom 4108 . . . . . . . . . . . 12 ((𝑊 × 𝑊) ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))}) = ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊))
15 inopab 5675 . . . . . . . . . . . 12 ({⟨𝑢, 𝑣⟩ ∣ (𝑢𝑊𝑣𝑊)} ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))}) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑊𝑣𝑊) ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
1613, 14, 153eqtr3i 2789 . . . . . . . . . . 11 ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊)) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑊𝑣𝑊) ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
17 vex 3413 . . . . . . . . . . . . . 14 𝑢 ∈ V
18 vex 3413 . . . . . . . . . . . . . 14 𝑣 ∈ V
1917, 18prss 4713 . . . . . . . . . . . . 13 ((𝑢𝑊𝑣𝑊) ↔ {𝑢, 𝑣} ⊆ 𝑊)
2019anbi1i 626 . . . . . . . . . . . 12 (((𝑢𝑊𝑣𝑊) ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))) ↔ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))))
2120opabbii 5102 . . . . . . . . . . 11 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑊𝑣𝑊) ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
2216, 21eqtri 2781 . . . . . . . . . 10 ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊)) = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
23 ereq1 8311 . . . . . . . . . 10 (({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊)) = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} → (({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊)) Er 𝑊 ↔ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} Er 𝑊))
2422, 23ax-mp 5 . . . . . . . . 9 (({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊)) Er 𝑊 ↔ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} Er 𝑊)
2511, 24sylib 221 . . . . . . . 8 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} Er 𝑊)
26 simplrl 776 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝑥𝑊)
27 fviss 6733 . . . . . . . . . . . . . . 15 ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
281, 27eqsstri 3928 . . . . . . . . . . . . . 14 𝑊 ⊆ Word (𝐼 × 2o)
2928, 26sseldi 3892 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝑥 ∈ Word (𝐼 × 2o))
30 opelxpi 5564 . . . . . . . . . . . . . . 15 ((𝑎𝐼𝑏 ∈ 2o) → ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2o))
3130adantl 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2o))
32 simprl 770 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝑎𝐼)
33 2oconcl 8143 . . . . . . . . . . . . . . . 16 (𝑏 ∈ 2o → (1o𝑏) ∈ 2o)
3433ad2antll 728 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (1o𝑏) ∈ 2o)
3532, 34opelxpd 5565 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ⟨𝑎, (1o𝑏)⟩ ∈ (𝐼 × 2o))
3631, 35s2cld 14285 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩ ∈ Word (𝐼 × 2o))
37 splcl 14166 . . . . . . . . . . . . 13 ((𝑥 ∈ Word (𝐼 × 2o) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩ ∈ Word (𝐼 × 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ∈ Word (𝐼 × 2o))
3829, 36, 37syl2anc 587 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ∈ Word (𝐼 × 2o))
391efgrcl 18913 . . . . . . . . . . . . . 14 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
4026, 39syl 17 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2o)))
4140simprd 499 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝑊 = Word (𝐼 × 2o))
4238, 41eleqtrrd 2855 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ∈ 𝑊)
43 pfxcl 14091 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ Word (𝐼 × 2o) → (𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o))
4429, 43syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o))
45 frgpup.b . . . . . . . . . . . . . . . . . . 19 𝐵 = (Base‘𝐻)
46 frgpup.n . . . . . . . . . . . . . . . . . . 19 𝑁 = (invg𝐻)
47 frgpup.t . . . . . . . . . . . . . . . . . . 19 𝑇 = (𝑦𝐼, 𝑧 ∈ 2o ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
48 frgpup.h . . . . . . . . . . . . . . . . . . 19 (𝜑𝐻 ∈ Grp)
49 frgpup.i . . . . . . . . . . . . . . . . . . 19 (𝜑𝐼𝑉)
50 frgpup.a . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:𝐼𝐵)
5145, 46, 47, 48, 49, 50frgpuptf 18968 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇:(𝐼 × 2o)⟶𝐵)
5251ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝑇:(𝐼 × 2o)⟶𝐵)
53 ccatco 14249 . . . . . . . . . . . . . . . . 17 (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩ ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)))
5444, 36, 52, 53syl3anc 1368 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)))
5554oveq2d 7171 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩))) = (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩))))
5648ad2antrr 725 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝐻 ∈ Grp)
57 grpmnd 18181 . . . . . . . . . . . . . . . . 17 (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
5856, 57syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝐻 ∈ Mnd)
59 wrdco 14245 . . . . . . . . . . . . . . . . 17 (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵)
6044, 52, 59syl2anc 587 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵)
61 wrdco 14245 . . . . . . . . . . . . . . . . 17 ((⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩ ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ∈ Word 𝐵)
6236, 52, 61syl2anc 587 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ∈ Word 𝐵)
63 eqid 2758 . . . . . . . . . . . . . . . . 17 (+g𝐻) = (+g𝐻)
6445, 63gsumccat 18077 . . . . . . . . . . . . . . . 16 ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵 ∧ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩))))
6558, 60, 62, 64syl3anc 1368 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩))))
6652, 31, 35s2co 14334 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) = ⟨“(𝑇‘⟨𝑎, 𝑏⟩)(𝑇‘⟨𝑎, (1o𝑏)⟩)”⟩)
67 df-ov 7158 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩)
6867a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩))
6967fveq2i 6665 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩))
70 df-ov 7158 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎(𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)𝑏) = ((𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)‘⟨𝑎, 𝑏⟩)
71 eqid 2758 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩) = (𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)
7271efgmval 18910 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝐼𝑏 ∈ 2o) → (𝑎(𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)𝑏) = ⟨𝑎, (1o𝑏)⟩)
7370, 72syl5eqr 2807 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎𝐼𝑏 ∈ 2o) → ((𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)‘⟨𝑎, 𝑏⟩) = ⟨𝑎, (1o𝑏)⟩)
7473adantl 485 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ((𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)‘⟨𝑎, 𝑏⟩) = ⟨𝑎, (1o𝑏)⟩)
7574fveq2d 6666 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇‘((𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑇‘⟨𝑎, (1o𝑏)⟩))
7645, 46, 47, 48, 49, 50, 71frgpuptinv 18969 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2o)) → (𝑇‘((𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
7730, 76sylan2 595 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇‘((𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
7877adantlr 714 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇‘((𝑦𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
7975, 78eqtr3d 2795 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇‘⟨𝑎, (1o𝑏)⟩) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
8069, 79eqtr4id 2812 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑁‘(𝑎𝑇𝑏)) = (𝑇‘⟨𝑎, (1o𝑏)⟩))
8168, 80s2eqd 14277 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩ = ⟨“(𝑇‘⟨𝑎, 𝑏⟩)(𝑇‘⟨𝑎, (1o𝑏)⟩)”⟩)
8266, 81eqtr4d 2796 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) = ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩)
8382oveq2d 7171 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) = (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩))
84 simprr 772 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝑏 ∈ 2o)
8552, 32, 84fovrnd 7321 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑎𝑇𝑏) ∈ 𝐵)
8645, 46grpinvcl 18223 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
8756, 85, 86syl2anc 587 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
8845, 63gsumws2 18078 . . . . . . . . . . . . . . . . . . 19 ((𝐻 ∈ Mnd ∧ (𝑎𝑇𝑏) ∈ 𝐵 ∧ (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g𝐻)(𝑁‘(𝑎𝑇𝑏))))
8958, 85, 87, 88syl3anc 1368 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g𝐻)(𝑁‘(𝑎𝑇𝑏))))
90 eqid 2758 . . . . . . . . . . . . . . . . . . . 20 (0g𝐻) = (0g𝐻)
9145, 63, 90, 46grprinv 18225 . . . . . . . . . . . . . . . . . . 19 ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → ((𝑎𝑇𝑏)(+g𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g𝐻))
9256, 85, 91syl2anc 587 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ((𝑎𝑇𝑏)(+g𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g𝐻))
9383, 89, 923eqtrd 2797 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) = (0g𝐻))
9493oveq2d 7171 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(0g𝐻)))
9545gsumwcl 18074 . . . . . . . . . . . . . . . . . 18 ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵)
9658, 60, 95syl2anc 587 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵)
9745, 63, 90grprid 18206 . . . . . . . . . . . . . . . . 17 ((𝐻 ∈ Grp ∧ (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) ∈ 𝐵) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(0g𝐻)) = (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))))
9856, 96, 97syl2anc 587 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(0g𝐻)) = (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))))
9994, 98eqtrd 2793 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩))) = (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))))
10055, 65, 993eqtrrd 2798 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛))) = (𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩))))
101100oveq1d 7170 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
102 swrdcl 14059 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word (𝐼 × 2o) → (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o))
10329, 102syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o))
104 wrdco 14245 . . . . . . . . . . . . . . 15 (((𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵)
105103, 52, 104syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵)
10645, 63gsumccat 18077 . . . . . . . . . . . . . 14 ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 prefix 𝑛)) ∈ Word 𝐵 ∧ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
10758, 60, 105, 106syl3anc 1368 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ (𝑥 prefix 𝑛)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
108 ccatcl 13978 . . . . . . . . . . . . . . . 16 (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩ ∈ Word (𝐼 × 2o)) → ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ∈ Word (𝐼 × 2o))
10944, 36, 108syl2anc 587 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ∈ Word (𝐼 × 2o))
110 wrdco 14245 . . . . . . . . . . . . . . 15 ((((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ∈ Word 𝐵)
111109, 52, 110syl2anc 587 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ∈ Word 𝐵)
11245, 63gsumccat 18077 . . . . . . . . . . . . . 14 ((𝐻 ∈ Mnd ∧ (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ∈ Word 𝐵 ∧ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
11358, 111, 105, 112syl3anc 1368 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
114101, 107, 1133eqtr4d 2803 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))) = (𝐻 Σg ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
115 simplrr 777 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝑛 ∈ (0...(♯‘𝑥)))
116 lencl 13937 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ Word (𝐼 × 2o) → (♯‘𝑥) ∈ ℕ0)
11729, 116syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (♯‘𝑥) ∈ ℕ0)
118 nn0uz 12325 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
119117, 118eleqtrdi 2862 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (♯‘𝑥) ∈ (ℤ‘0))
120 eluzfz2 12969 . . . . . . . . . . . . . . . . . 18 ((♯‘𝑥) ∈ (ℤ‘0) → (♯‘𝑥) ∈ (0...(♯‘𝑥)))
121119, 120syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (♯‘𝑥) ∈ (0...(♯‘𝑥)))
122 ccatpfx 14115 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ Word (𝐼 × 2o) ∧ 𝑛 ∈ (0...(♯‘𝑥)) ∧ (♯‘𝑥) ∈ (0...(♯‘𝑥))) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = (𝑥 prefix (♯‘𝑥)))
12329, 115, 121, 122syl3anc 1368 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = (𝑥 prefix (♯‘𝑥)))
124 pfxid 14098 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ Word (𝐼 × 2o) → (𝑥 prefix (♯‘𝑥)) = 𝑥)
12529, 124syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑥 prefix (♯‘𝑥)) = 𝑥)
126123, 125eqtrd 2793 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)) = 𝑥)
127126coeq2d 5707 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = (𝑇𝑥))
128 ccatco 14249 . . . . . . . . . . . . . . 15 (((𝑥 prefix 𝑛) ∈ Word (𝐼 × 2o) ∧ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
12944, 103, 52, 128syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ ((𝑥 prefix 𝑛) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
130127, 129eqtr3d 2795 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇𝑥) = ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
131130oveq2d 7171 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg (𝑇𝑥)) = (𝐻 Σg ((𝑇 ∘ (𝑥 prefix 𝑛)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
132 splval 14165 . . . . . . . . . . . . . . . 16 ((𝑥𝑊 ∧ (𝑛 ∈ (0...(♯‘𝑥)) ∧ 𝑛 ∈ (0...(♯‘𝑥)) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩ ∈ Word (𝐼 × 2o))) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) = (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))
13326, 115, 115, 36, 132syl13anc 1369 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) = (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))
134133coeq2d 5707 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) = (𝑇 ∘ (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
135 ccatco 14249 . . . . . . . . . . . . . . 15 ((((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ∈ Word (𝐼 × 2o) ∧ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩) ∈ Word (𝐼 × 2o) ∧ 𝑇:(𝐼 × 2o)⟶𝐵) → (𝑇 ∘ (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
136109, 103, 52, 135syl3anc 1368 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ (((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))) = ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
137134, 136eqtrd 2793 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) = ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩))))
138137oveq2d 7171 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))) = (𝐻 Σg ((𝑇 ∘ ((𝑥 prefix 𝑛) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (♯‘𝑥)⟩)))))
139114, 131, 1383eqtr4d 2803 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → (𝐻 Σg (𝑇𝑥)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))))
140 vex 3413 . . . . . . . . . . . 12 𝑥 ∈ V
141 ovex 7188 . . . . . . . . . . . 12 (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ∈ V
142 eleq1 2839 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → (𝑢𝑊𝑥𝑊))
143 eleq1 2839 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) → (𝑣𝑊 ↔ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ∈ 𝑊))
144142, 143bi2anan9 638 . . . . . . . . . . . . . 14 ((𝑢 = 𝑥𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) → ((𝑢𝑊𝑣𝑊) ↔ (𝑥𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ∈ 𝑊)))
14519, 144bitr3id 288 . . . . . . . . . . . . 13 ((𝑢 = 𝑥𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) → ({𝑢, 𝑣} ⊆ 𝑊 ↔ (𝑥𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ∈ 𝑊)))
146 coeq2 5703 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → (𝑇𝑢) = (𝑇𝑥))
147146oveq2d 7171 . . . . . . . . . . . . . 14 (𝑢 = 𝑥 → (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑥)))
148 coeq2 5703 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) → (𝑇𝑣) = (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
149148oveq2d 7171 . . . . . . . . . . . . . 14 (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) → (𝐻 Σg (𝑇𝑣)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))))
150147, 149eqeqan12d 2775 . . . . . . . . . . . . 13 ((𝑢 = 𝑥𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) → ((𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)) ↔ (𝐻 Σg (𝑇𝑥)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))))
151145, 150anbi12d 633 . . . . . . . . . . . 12 ((𝑢 = 𝑥𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) → (({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))) ↔ ((𝑥𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ∈ 𝑊) ∧ (𝐻 Σg (𝑇𝑥)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))))))
152 eqid 2758 . . . . . . . . . . . 12 {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
153140, 141, 151, 152braba 5397 . . . . . . . . . . 11 (𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ ((𝑥𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ∈ 𝑊) ∧ (𝐻 Σg (𝑇𝑥)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))))
15426, 42, 139, 153syl21anbrc 1341 . . . . . . . . . 10 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2o)) → 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
155154ralrimivva 3120 . . . . . . . . 9 ((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(♯‘𝑥)))) → ∀𝑎𝐼𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
156155ralrimivva 3120 . . . . . . . 8 (𝜑 → ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))
1571fvexi 6676 . . . . . . . . . 10 𝑊 ∈ V
158 erex 8328 . . . . . . . . . 10 ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} Er 𝑊 → (𝑊 ∈ V → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} ∈ V))
15925, 157, 158mpisyl 21 . . . . . . . . 9 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} ∈ V)
160 ereq1 8311 . . . . . . . . . . 11 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} → (𝑟 Er 𝑊 ↔ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} Er 𝑊))
161 breq 5037 . . . . . . . . . . . . 13 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} → (𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
1621612ralbidv 3128 . . . . . . . . . . . 12 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} → (∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ ∀𝑎𝐼𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
1631622ralbidv 3128 . . . . . . . . . . 11 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} → (∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩) ↔ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)))
164160, 163anbi12d 633 . . . . . . . . . 10 (𝑟 = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} → ((𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩)) ↔ ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))))
165164elabg 3589 . . . . . . . . 9 ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} ∈ V → ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))} ↔ ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))))
166159, 165syl 17 . . . . . . . 8 (𝜑 → ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))} ↔ ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))))
16725, 156, 166mpbir2and 712 . . . . . . 7 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))})
168 intss1 4856 . . . . . . 7 ({⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))} → {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))} ⊆ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))})
169167, 168syl 17 . . . . . 6 (𝜑 {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(♯‘𝑥))∀𝑎𝐼𝑏 ∈ 2o 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1o𝑏)⟩”⟩⟩))} ⊆ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))})
1703, 169eqsstrid 3942 . . . . 5 (𝜑 ⊆ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))})
171170ssbrd 5078 . . . 4 (𝜑 → (𝐴 𝐶𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}𝐶))
172171imp 410 . . 3 ((𝜑𝐴 𝐶) → 𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}𝐶)
1731, 2efger 18916 . . . . . 6 Er 𝑊
174 errel 8313 . . . . . 6 ( Er 𝑊 → Rel )
175173, 174mp1i 13 . . . . 5 (𝜑 → Rel )
176 brrelex12 5577 . . . . 5 ((Rel 𝐴 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
177175, 176sylan 583 . . . 4 ((𝜑𝐴 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
178 preq12 4631 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐶) → {𝑢, 𝑣} = {𝐴, 𝐶})
179178sseq1d 3925 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐶) → ({𝑢, 𝑣} ⊆ 𝑊 ↔ {𝐴, 𝐶} ⊆ 𝑊))
180 coeq2 5703 . . . . . . . 8 (𝑢 = 𝐴 → (𝑇𝑢) = (𝑇𝐴))
181180oveq2d 7171 . . . . . . 7 (𝑢 = 𝐴 → (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝐴)))
182 coeq2 5703 . . . . . . . 8 (𝑣 = 𝐶 → (𝑇𝑣) = (𝑇𝐶))
183182oveq2d 7171 . . . . . . 7 (𝑣 = 𝐶 → (𝐻 Σg (𝑇𝑣)) = (𝐻 Σg (𝑇𝐶)))
184181, 183eqeqan12d 2775 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐶) → ((𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)) ↔ (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶))))
185179, 184anbi12d 633 . . . . 5 ((𝑢 = 𝐴𝑣 = 𝐶) → (({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))) ↔ ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))))
186185, 152brabga 5394 . . . 4 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}𝐶 ↔ ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))))
187177, 186syl 17 . . 3 ((𝜑𝐴 𝐶) → (𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}𝐶 ↔ ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))))
188172, 187mpbid 235 . 2 ((𝜑𝐴 𝐶) → ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶))))
189188simprd 499 1 ((𝜑𝐴 𝐶) → (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2735  ∀wral 3070  Vcvv 3409   ∖ cdif 3857   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ifcif 4423  {cpr 4527  ⟨cop 4531  ⟨cotp 4533  ∩ cint 4841   class class class wbr 5035  {copab 5097   I cid 5432   × cxp 5525   ∘ ccom 5531  Rel wrel 5532  ⟶wf 6335  ‘cfv 6339  (class class class)co 7155   ∈ cmpo 7157  1oc1o 8110  2oc2o 8111   Er wer 8301  0cc0 10580  ℕ0cn0 11939  ℤ≥cuz 12287  ...cfz 12944  ♯chash 13745  Word cword 13918   ++ cconcat 13974   substr csubstr 14054   prefix cpfx 14084   splice csplice 14163  ⟨“cs2 14255  Basecbs 16546  +gcplusg 16628  0gc0g 16776   Σg cgsu 16777  Mndcmnd 17982  Grpcgrp 18174  invgcminusg 18175   ~FG cefg 18904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-ot 4534  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-1o 8117  df-2o 8118  df-er 8304  df-map 8423  df-en 8533  df-dom 8534  df-sdom 8535  df-fin 8536  df-card 9406  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-n0 11940  df-z 12026  df-uz 12288  df-fz 12945  df-fzo 13088  df-seq 13424  df-hash 13746  df-word 13919  df-concat 13975  df-s1 14002  df-substr 14055  df-pfx 14085  df-splice 14164  df-s2 14262  df-ndx 16549  df-slot 16550  df-base 16552  df-sets 16553  df-ress 16554  df-plusg 16641  df-0g 16778  df-gsum 16779  df-mgm 17923  df-sgrp 17972  df-mnd 17983  df-submnd 18028  df-grp 18177  df-minusg 18178  df-efg 18907 This theorem is referenced by:  frgpupf  18971
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