MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frgpuplem Structured version   Visualization version   GIF version

Theorem frgpuplem 18231
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 18176 . . . . . 6 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑥𝑊𝑛 ∈ (0...(#‘𝑥))∀𝑎𝐼𝑏 ∈ 2𝑜 𝑥𝑟(𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))}
4 coeq2 5313 . . . . . . . . . . . . 13 (𝑢 = 𝑣 → (𝑇𝑢) = (𝑇𝑣))
54oveq2d 6706 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))
6 eqid 2651 . . . . . . . . . . . 12 {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} = {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))}
75, 6eqer 7822 . . . . . . . . . . 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 3658 . . . . . . . . . . 11 𝑊 ⊆ V
109a1i 11 . . . . . . . . . 10 (𝜑𝑊 ⊆ V)
118, 10erinxp 7864 . . . . . . . . 9 (𝜑 → ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊)) Er 𝑊)
12 df-xp 5149 . . . . . . . . . . . . 13 (𝑊 × 𝑊) = {⟨𝑢, 𝑣⟩ ∣ (𝑢𝑊𝑣𝑊)}
1312ineq1i 3843 . . . . . . . . . . . 12 ((𝑊 × 𝑊) ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))}) = ({⟨𝑢, 𝑣⟩ ∣ (𝑢𝑊𝑣𝑊)} ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))})
14 incom 3838 . . . . . . . . . . . 12 ((𝑊 × 𝑊) ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))}) = ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊))
15 inopab 5285 . . . . . . . . . . . 12 ({⟨𝑢, 𝑣⟩ ∣ (𝑢𝑊𝑣𝑊)} ∩ {⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))}) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑊𝑣𝑊) ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
1613, 14, 153eqtr3i 2681 . . . . . . . . . . 11 ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊)) = {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑊𝑣𝑊) ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
17 vex 3234 . . . . . . . . . . . . . 14 𝑢 ∈ V
18 vex 3234 . . . . . . . . . . . . . 14 𝑣 ∈ V
1917, 18prss 4383 . . . . . . . . . . . . 13 ((𝑢𝑊𝑣𝑊) ↔ {𝑢, 𝑣} ⊆ 𝑊)
2019anbi1i 731 . . . . . . . . . . . 12 (((𝑢𝑊𝑣𝑊) ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))) ↔ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))))
2120opabbii 4750 . . . . . . . . . . 11 {⟨𝑢, 𝑣⟩ ∣ ((𝑢𝑊𝑣𝑊) ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
2216, 21eqtri 2673 . . . . . . . . . 10 ({⟨𝑢, 𝑣⟩ ∣ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))} ∩ (𝑊 × 𝑊)) = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
23 ereq1 7794 . . . . . . . . . 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 208 . . . . . . . 8 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} Er 𝑊)
26 simplrl 817 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 𝑥𝑊)
27 fviss 6295 . . . . . . . . . . . . . . . 16 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
281, 27eqsstri 3668 . . . . . . . . . . . . . . 15 𝑊 ⊆ Word (𝐼 × 2𝑜)
2928, 26sseldi 3634 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 𝑥 ∈ Word (𝐼 × 2𝑜))
30 opelxpi 5182 . . . . . . . . . . . . . . . 16 ((𝑎𝐼𝑏 ∈ 2𝑜) → ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2𝑜))
3130adantl 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2𝑜))
32 simprl 809 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 𝑎𝐼)
33 2oconcl 7628 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ 2𝑜 → (1𝑜𝑏) ∈ 2𝑜)
3433ad2antll 765 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (1𝑜𝑏) ∈ 2𝑜)
35 opelxpi 5182 . . . . . . . . . . . . . . . 16 ((𝑎𝐼 ∧ (1𝑜𝑏) ∈ 2𝑜) → ⟨𝑎, (1𝑜𝑏)⟩ ∈ (𝐼 × 2𝑜))
3632, 34, 35syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ⟨𝑎, (1𝑜𝑏)⟩ ∈ (𝐼 × 2𝑜))
3731, 36s2cld 13662 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ ∈ Word (𝐼 × 2𝑜))
38 splcl 13549 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (𝐼 × 2𝑜) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ ∈ Word (𝐼 × 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ∈ Word (𝐼 × 2𝑜))
3929, 37, 38syl2anc 694 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ∈ Word (𝐼 × 2𝑜))
401efgrcl 18174 . . . . . . . . . . . . . . 15 (𝑥𝑊 → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
4126, 40syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐼 ∈ V ∧ 𝑊 = Word (𝐼 × 2𝑜)))
4241simprd 478 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 𝑊 = Word (𝐼 × 2𝑜))
4339, 42eleqtrrd 2733 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ∈ 𝑊)
4426, 43jca 553 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑥𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ∈ 𝑊))
45 swrdcl 13464 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ Word (𝐼 × 2𝑜) → (𝑥 substr ⟨0, 𝑛⟩) ∈ Word (𝐼 × 2𝑜))
4629, 45syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑥 substr ⟨0, 𝑛⟩) ∈ Word (𝐼 × 2𝑜))
47 frgpup.b . . . . . . . . . . . . . . . . . . 19 𝐵 = (Base‘𝐻)
48 frgpup.n . . . . . . . . . . . . . . . . . . 19 𝑁 = (invg𝐻)
49 frgpup.t . . . . . . . . . . . . . . . . . . 19 𝑇 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ if(𝑧 = ∅, (𝐹𝑦), (𝑁‘(𝐹𝑦))))
50 frgpup.h . . . . . . . . . . . . . . . . . . 19 (𝜑𝐻 ∈ Grp)
51 frgpup.i . . . . . . . . . . . . . . . . . . 19 (𝜑𝐼𝑉)
52 frgpup.a . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:𝐼𝐵)
5347, 48, 49, 50, 51, 52frgpuptf 18229 . . . . . . . . . . . . . . . . . 18 (𝜑𝑇:(𝐼 × 2𝑜)⟶𝐵)
5453ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 𝑇:(𝐼 × 2𝑜)⟶𝐵)
55 ccatco 13627 . . . . . . . . . . . . . . . . 17 (((𝑥 substr ⟨0, 𝑛⟩) ∈ Word (𝐼 × 2𝑜) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) = ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)))
5646, 37, 54, 55syl3anc 1366 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) = ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)))
5756oveq2d 6706 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩))) = (𝐻 Σg ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩))))
5850ad2antrr 762 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 𝐻 ∈ Grp)
59 grpmnd 17476 . . . . . . . . . . . . . . . . 17 (𝐻 ∈ Grp → 𝐻 ∈ Mnd)
6058, 59syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 𝐻 ∈ Mnd)
61 wrdco 13623 . . . . . . . . . . . . . . . . 17 (((𝑥 substr ⟨0, 𝑛⟩) ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ∈ Word 𝐵)
6246, 54, 61syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ∈ Word 𝐵)
63 wrdco 13623 . . . . . . . . . . . . . . . . 17 ((⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ∈ Word 𝐵)
6437, 54, 63syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ∈ Word 𝐵)
65 eqid 2651 . . . . . . . . . . . . . . . . 17 (+g𝐻) = (+g𝐻)
6647, 65gsumccat 17425 . . . . . . . . . . . . . . . 16 ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ∈ Word 𝐵 ∧ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩))) = ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩))))
6760, 62, 64, 66syl3anc 1366 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩))) = ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩))))
6854, 31, 36s2co 13711 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) = ⟨“(𝑇‘⟨𝑎, 𝑏⟩)(𝑇‘⟨𝑎, (1𝑜𝑏)⟩)”⟩)
69 df-ov 6693 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩)
7069a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑎𝑇𝑏) = (𝑇‘⟨𝑎, 𝑏⟩))
71 df-ov 6693 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎(𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)𝑏) = ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘⟨𝑎, 𝑏⟩)
72 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩) = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
7372efgmval 18171 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑎𝐼𝑏 ∈ 2𝑜) → (𝑎(𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)𝑏) = ⟨𝑎, (1𝑜𝑏)⟩)
7471, 73syl5eqr 2699 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑎𝐼𝑏 ∈ 2𝑜) → ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘⟨𝑎, 𝑏⟩) = ⟨𝑎, (1𝑜𝑏)⟩)
7574adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘⟨𝑎, 𝑏⟩) = ⟨𝑎, (1𝑜𝑏)⟩)
7675fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇‘((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑇‘⟨𝑎, (1𝑜𝑏)⟩))
7747, 48, 49, 50, 51, 52, 72frgpuptinv 18230 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ ⟨𝑎, 𝑏⟩ ∈ (𝐼 × 2𝑜)) → (𝑇‘((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
7830, 77sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇‘((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
7978adantlr 751 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇‘((𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)‘⟨𝑎, 𝑏⟩)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
8076, 79eqtr3d 2687 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇‘⟨𝑎, (1𝑜𝑏)⟩) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩)))
8169fveq2i 6232 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁‘(𝑎𝑇𝑏)) = (𝑁‘(𝑇‘⟨𝑎, 𝑏⟩))
8280, 81syl6reqr 2704 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑁‘(𝑎𝑇𝑏)) = (𝑇‘⟨𝑎, (1𝑜𝑏)⟩))
8370, 82s2eqd 13654 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩ = ⟨“(𝑇‘⟨𝑎, 𝑏⟩)(𝑇‘⟨𝑎, (1𝑜𝑏)⟩)”⟩)
8468, 83eqtr4d 2688 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) = ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩)
8584oveq2d 6706 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) = (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩))
86 simprr 811 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 𝑏 ∈ 2𝑜)
8754, 32, 86fovrnd 6848 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑎𝑇𝑏) ∈ 𝐵)
8847, 48grpinvcl 17514 . . . . . . . . . . . . . . . . . . . 20 ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
8958, 87, 88syl2anc 694 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵)
9047, 65gsumws2 17426 . . . . . . . . . . . . . . . . . . 19 ((𝐻 ∈ Mnd ∧ (𝑎𝑇𝑏) ∈ 𝐵 ∧ (𝑁‘(𝑎𝑇𝑏)) ∈ 𝐵) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g𝐻)(𝑁‘(𝑎𝑇𝑏))))
9160, 87, 89, 90syl3anc 1366 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg ⟨“(𝑎𝑇𝑏)(𝑁‘(𝑎𝑇𝑏))”⟩) = ((𝑎𝑇𝑏)(+g𝐻)(𝑁‘(𝑎𝑇𝑏))))
92 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 (0g𝐻) = (0g𝐻)
9347, 65, 92, 48grprinv 17516 . . . . . . . . . . . . . . . . . . 19 ((𝐻 ∈ Grp ∧ (𝑎𝑇𝑏) ∈ 𝐵) → ((𝑎𝑇𝑏)(+g𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g𝐻))
9458, 87, 93syl2anc 694 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ((𝑎𝑇𝑏)(+g𝐻)(𝑁‘(𝑎𝑇𝑏))) = (0g𝐻))
9585, 91, 943eqtrd 2689 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) = (0g𝐻))
9695oveq2d 6706 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩))) = ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(0g𝐻)))
9747gsumwcl 17424 . . . . . . . . . . . . . . . . . 18 ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ∈ Word 𝐵) → (𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩))) ∈ 𝐵)
9860, 62, 97syl2anc 694 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩))) ∈ 𝐵)
9947, 65, 92grprid 17500 . . . . . . . . . . . . . . . . 17 ((𝐻 ∈ Grp ∧ (𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩))) ∈ 𝐵) → ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(0g𝐻)) = (𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩))))
10058, 98, 99syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(0g𝐻)) = (𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩))))
10196, 100eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩))) = (𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩))))
10257, 67, 1013eqtrrd 2690 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩))) = (𝐻 Σg (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩))))
103102oveq1d 6705 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))))
104 swrdcl 13464 . . . . . . . . . . . . . . . 16 (𝑥 ∈ Word (𝐼 × 2𝑜) → (𝑥 substr ⟨𝑛, (#‘𝑥)⟩) ∈ Word (𝐼 × 2𝑜))
10529, 104syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑥 substr ⟨𝑛, (#‘𝑥)⟩) ∈ Word (𝐼 × 2𝑜))
106 wrdco 13623 . . . . . . . . . . . . . . 15 (((𝑥 substr ⟨𝑛, (#‘𝑥)⟩) ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)) ∈ Word 𝐵)
107105, 54, 106syl2anc 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)) ∈ Word 𝐵)
10847, 65gsumccat 17425 . . . . . . . . . . . . . 14 ((𝐻 ∈ Mnd ∧ (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ∈ Word 𝐵 ∧ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))))
10960, 62, 107, 108syl3anc 1366 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))))
110 ccatcl 13392 . . . . . . . . . . . . . . . 16 (((𝑥 substr ⟨0, 𝑛⟩) ∈ Word (𝐼 × 2𝑜) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ ∈ Word (𝐼 × 2𝑜)) → ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ∈ Word (𝐼 × 2𝑜))
11146, 37, 110syl2anc 694 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ∈ Word (𝐼 × 2𝑜))
112 wrdco 13623 . . . . . . . . . . . . . . 15 ((((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ∈ Word 𝐵)
113111, 54, 112syl2anc 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ∈ Word 𝐵)
11447, 65gsumccat 17425 . . . . . . . . . . . . . 14 ((𝐻 ∈ Mnd ∧ (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ∈ Word 𝐵 ∧ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)) ∈ Word 𝐵) → (𝐻 Σg ((𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))))
11560, 113, 107, 114syl3anc 1366 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg ((𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))) = ((𝐻 Σg (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)))(+g𝐻)(𝐻 Σg (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))))
116103, 109, 1153eqtr4d 2695 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))) = (𝐻 Σg ((𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))))
117 simplrr 818 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 𝑛 ∈ (0...(#‘𝑥)))
118 elfzuz 12376 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (0...(#‘𝑥)) → 𝑛 ∈ (ℤ‘0))
119 eluzfz1 12386 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (ℤ‘0) → 0 ∈ (0...𝑛))
120117, 118, 1193syl 18 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → 0 ∈ (0...𝑛))
121 lencl 13356 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ Word (𝐼 × 2𝑜) → (#‘𝑥) ∈ ℕ0)
12229, 121syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (#‘𝑥) ∈ ℕ0)
123 nn0uz 11760 . . . . . . . . . . . . . . . . . . 19 0 = (ℤ‘0)
124122, 123syl6eleq 2740 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (#‘𝑥) ∈ (ℤ‘0))
125 eluzfz2 12387 . . . . . . . . . . . . . . . . . 18 ((#‘𝑥) ∈ (ℤ‘0) → (#‘𝑥) ∈ (0...(#‘𝑥)))
126124, 125syl 17 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (#‘𝑥) ∈ (0...(#‘𝑥)))
127 ccatswrd 13502 . . . . . . . . . . . . . . . . 17 ((𝑥 ∈ Word (𝐼 × 2𝑜) ∧ (0 ∈ (0...𝑛) ∧ 𝑛 ∈ (0...(#‘𝑥)) ∧ (#‘𝑥) ∈ (0...(#‘𝑥)))) → ((𝑥 substr ⟨0, 𝑛⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)) = (𝑥 substr ⟨0, (#‘𝑥)⟩))
12829, 120, 117, 126, 127syl13anc 1368 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ((𝑥 substr ⟨0, 𝑛⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)) = (𝑥 substr ⟨0, (#‘𝑥)⟩))
129 swrdid 13474 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ Word (𝐼 × 2𝑜) → (𝑥 substr ⟨0, (#‘𝑥)⟩) = 𝑥)
13029, 129syl 17 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑥 substr ⟨0, (#‘𝑥)⟩) = 𝑥)
131128, 130eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → ((𝑥 substr ⟨0, 𝑛⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)) = 𝑥)
132131coeq2d 5317 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))) = (𝑇𝑥))
133 ccatco 13627 . . . . . . . . . . . . . . 15 (((𝑥 substr ⟨0, 𝑛⟩) ∈ Word (𝐼 × 2𝑜) ∧ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩) ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))) = ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))))
13446, 105, 54, 133syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))) = ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))))
135132, 134eqtr3d 2687 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇𝑥) = ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))))
136135oveq2d 6706 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg (𝑇𝑥)) = (𝐻 Σg ((𝑇 ∘ (𝑥 substr ⟨0, 𝑛⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))))
137 splval 13548 . . . . . . . . . . . . . . . 16 ((𝑥𝑊 ∧ (𝑛 ∈ (0...(#‘𝑥)) ∧ 𝑛 ∈ (0...(#‘𝑥)) ∧ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩ ∈ Word (𝐼 × 2𝑜))) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))
13826, 117, 117, 37, 137syl13anc 1368 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) = (((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))
139138coeq2d 5317 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)) = (𝑇 ∘ (((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))))
140 ccatco 13627 . . . . . . . . . . . . . . 15 ((((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ∈ Word (𝐼 × 2𝑜) ∧ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩) ∈ Word (𝐼 × 2𝑜) ∧ 𝑇:(𝐼 × 2𝑜)⟶𝐵) → (𝑇 ∘ (((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))) = ((𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))))
141111, 105, 54, 140syl3anc 1366 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ (((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩) ++ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))) = ((𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))))
142139, 141eqtrd 2685 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)) = ((𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩))))
143142oveq2d 6706 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))) = (𝐻 Σg ((𝑇 ∘ ((𝑥 substr ⟨0, 𝑛⟩) ++ ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩)) ++ (𝑇 ∘ (𝑥 substr ⟨𝑛, (#‘𝑥)⟩)))))
144116, 136, 1433eqtr4d 2695 . . . . . . . . . . 11 (((𝜑 ∧ (𝑥𝑊𝑛 ∈ (0...(#‘𝑥)))) ∧ (𝑎𝐼𝑏 ∈ 2𝑜)) → (𝐻 Σg (𝑇𝑥)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))))
145 vex 3234 . . . . . . . . . . . 12 𝑥 ∈ V
146 ovex 6718 . . . . . . . . . . . 12 (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ∈ V
147 eleq1 2718 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → (𝑢𝑊𝑥𝑊))
148 eleq1 2718 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) → (𝑣𝑊 ↔ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ∈ 𝑊))
149147, 148bi2anan9 935 . . . . . . . . . . . . . 14 ((𝑢 = 𝑥𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)) → ((𝑢𝑊𝑣𝑊) ↔ (𝑥𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ∈ 𝑊)))
15019, 149syl5bbr 274 . . . . . . . . . . . . 13 ((𝑢 = 𝑥𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)) → ({𝑢, 𝑣} ⊆ 𝑊 ↔ (𝑥𝑊 ∧ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) ∈ 𝑊)))
151 coeq2 5313 . . . . . . . . . . . . . . 15 (𝑢 = 𝑥 → (𝑇𝑢) = (𝑇𝑥))
152151oveq2d 6706 . . . . . . . . . . . . . 14 (𝑢 = 𝑥 → (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑥)))
153 coeq2 5313 . . . . . . . . . . . . . . 15 (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) → (𝑇𝑣) = (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩)))
154153oveq2d 6706 . . . . . . . . . . . . . 14 (𝑣 = (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩) → (𝐻 Σg (𝑇𝑣)) = (𝐻 Σg (𝑇 ∘ (𝑥 splice ⟨𝑛, 𝑛, ⟨“⟨𝑎, 𝑏⟩⟨𝑎, (1𝑜𝑏)⟩”⟩⟩))))
155152, 154eqeqan12d 2667 . . . . . . . . . . . . 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)⟩”⟩⟩)))))
156150, 155anbi12d 747 . . . . . . . . . . . 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)⟩”⟩⟩))))))
157 eqid 2651 . . . . . . . . . . . 12 {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} = {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}
158145, 146, 156, 157braba 5021 . . . . . . . . . . 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)⟩”⟩⟩)))))
15944, 144, 158sylanbrc 699 . . . . . . . . . 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)⟩”⟩⟩))
160159ralrimivva 3000 . . . . . . . . 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)⟩”⟩⟩))
161160ralrimivva 3000 . . . . . . . 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)⟩”⟩⟩))
162 fvex 6239 . . . . . . . . . . 11 ( I ‘Word (𝐼 × 2𝑜)) ∈ V
1631, 162eqeltri 2726 . . . . . . . . . 10 𝑊 ∈ V
164 erex 7811 . . . . . . . . . 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))
16525, 163, 164mpisyl 21 . . . . . . . . 9 (𝜑 → {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))} ∈ V)
166 ereq1 7794 . . . . . . . . . . 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))
167 breq 4687 . . . . . . . . . . . . 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)⟩”⟩⟩)))
1681672ralbidv 3018 . . . . . . . . . . . 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)⟩”⟩⟩)))
1691682ralbidv 3018 . . . . . . . . . . 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)⟩”⟩⟩)))
170166, 169anbi12d 747 . . . . . . . . . 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)⟩”⟩⟩))))
171170elabg 3383 . . . . . . . . 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)⟩”⟩⟩))))
172165, 171syl 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)⟩”⟩⟩))))
17325, 161, 172mpbir2and 977 . . . . . . 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)⟩”⟩⟩))})
174 intss1 4524 . . . . . . 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)))})
175173, 174syl 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)))})
1763, 175syl5eqss 3682 . . . . 5 (𝜑 ⊆ {⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))})
177176ssbrd 4728 . . . 4 (𝜑 → (𝐴 𝐶𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}𝐶))
178177imp 444 . . 3 ((𝜑𝐴 𝐶) → 𝐴{⟨𝑢, 𝑣⟩ ∣ ({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)))}𝐶)
1791, 2efger 18177 . . . . . 6 Er 𝑊
180 errel 7796 . . . . . 6 ( Er 𝑊 → Rel )
181179, 180mp1i 13 . . . . 5 (𝜑 → Rel )
182 brrelex12 5189 . . . . 5 ((Rel 𝐴 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
183181, 182sylan 487 . . . 4 ((𝜑𝐴 𝐶) → (𝐴 ∈ V ∧ 𝐶 ∈ V))
184 preq12 4302 . . . . . . 7 ((𝑢 = 𝐴𝑣 = 𝐶) → {𝑢, 𝑣} = {𝐴, 𝐶})
185184sseq1d 3665 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐶) → ({𝑢, 𝑣} ⊆ 𝑊 ↔ {𝐴, 𝐶} ⊆ 𝑊))
186 coeq2 5313 . . . . . . . 8 (𝑢 = 𝐴 → (𝑇𝑢) = (𝑇𝐴))
187186oveq2d 6706 . . . . . . 7 (𝑢 = 𝐴 → (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝐴)))
188 coeq2 5313 . . . . . . . 8 (𝑣 = 𝐶 → (𝑇𝑣) = (𝑇𝐶))
189188oveq2d 6706 . . . . . . 7 (𝑣 = 𝐶 → (𝐻 Σg (𝑇𝑣)) = (𝐻 Σg (𝑇𝐶)))
190187, 189eqeqan12d 2667 . . . . . 6 ((𝑢 = 𝐴𝑣 = 𝐶) → ((𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣)) ↔ (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶))))
191185, 190anbi12d 747 . . . . 5 ((𝑢 = 𝐴𝑣 = 𝐶) → (({𝑢, 𝑣} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝑢)) = (𝐻 Σg (𝑇𝑣))) ↔ ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))))
192191, 157brabga 5018 . . . 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)))))
193183, 192syl 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)))))
194178, 193mpbid 222 . 2 ((𝜑𝐴 𝐶) → ({𝐴, 𝐶} ⊆ 𝑊 ∧ (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶))))
195194simprd 478 1 ((𝜑𝐴 𝐶) → (𝐻 Σg (𝑇𝐴)) = (𝐻 Σg (𝑇𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  ifcif 4119  {cpr 4212  cop 4216  cotp 4218   cint 4507   class class class wbr 4685  {copab 4745   I cid 5052   × cxp 5141  ccom 5147  Rel wrel 5148  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1𝑜c1o 7598  2𝑜c2o 7599   Er wer 7784  0cc0 9974  0cn0 11330  cuz 11725  ...cfz 12364  #chash 13157  Word cword 13323   ++ cconcat 13325   substr csubstr 13327   splice csplice 13328  ⟨“cs2 13632  Basecbs 15904  +gcplusg 15988  0gc0g 16147   Σg cgsu 16148  Mndcmnd 17341  Grpcgrp 17469  invgcminusg 17470   ~FG cefg 18165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-substr 13335  df-splice 13336  df-s2 13639  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-efg 18168
This theorem is referenced by:  frgpupf  18232
  Copyright terms: Public domain W3C validator