| Step | Hyp | Ref
| Expression |
| 1 | | noel 4338 |
. . . 4
⊢ ¬
𝐴 ∈
∅ |
| 2 | | psgnunilem2.id |
. . . . . . . 8
⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) |
| 3 | 2 | difeq1d 4125 |
. . . . . . 7
⊢ (𝜑 → ((𝐺 Σg 𝑊) ∖ I ) = (( I ↾
𝐷) ∖ I
)) |
| 4 | 3 | dmeqd 5916 |
. . . . . 6
⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (( I ↾
𝐷) ∖ I
)) |
| 5 | | resss 6019 |
. . . . . . . . 9
⊢ ( I
↾ 𝐷) ⊆
I |
| 6 | | ssdif0 4366 |
. . . . . . . . 9
⊢ (( I
↾ 𝐷) ⊆ I ↔
(( I ↾ 𝐷) ∖ I )
= ∅) |
| 7 | 5, 6 | mpbi 230 |
. . . . . . . 8
⊢ (( I
↾ 𝐷) ∖ I ) =
∅ |
| 8 | 7 | dmeqi 5915 |
. . . . . . 7
⊢ dom (( I
↾ 𝐷) ∖ I ) =
dom ∅ |
| 9 | | dm0 5931 |
. . . . . . 7
⊢ dom
∅ = ∅ |
| 10 | 8, 9 | eqtri 2765 |
. . . . . 6
⊢ dom (( I
↾ 𝐷) ∖ I ) =
∅ |
| 11 | 4, 10 | eqtrdi 2793 |
. . . . 5
⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) =
∅) |
| 12 | 11 | eleq2d 2827 |
. . . 4
⊢ (𝜑 → (𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I ) ↔ 𝐴 ∈
∅)) |
| 13 | 1, 12 | mtbiri 327 |
. . 3
⊢ (𝜑 → ¬ 𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I )) |
| 14 | | psgnunilem2.d |
. . . . . . . . 9
⊢ (𝜑 → 𝐷 ∈ 𝑉) |
| 15 | | psgnunilem2.g |
. . . . . . . . . 10
⊢ 𝐺 = (SymGrp‘𝐷) |
| 16 | 15 | symggrp 19418 |
. . . . . . . . 9
⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp) |
| 17 | | grpmnd 18958 |
. . . . . . . . 9
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
| 18 | 14, 16, 17 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ Mnd) |
| 19 | | psgnunilem2.t |
. . . . . . . . . . . 12
⊢ 𝑇 = ran (pmTrsp‘𝐷) |
| 20 | | eqid 2737 |
. . . . . . . . . . . 12
⊢
(Base‘𝐺) =
(Base‘𝐺) |
| 21 | 19, 15, 20 | symgtrf 19487 |
. . . . . . . . . . 11
⊢ 𝑇 ⊆ (Base‘𝐺) |
| 22 | | sswrd 14560 |
. . . . . . . . . . 11
⊢ (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺)) |
| 23 | 21, 22 | mp1i 13 |
. . . . . . . . . 10
⊢ (𝜑 → Word 𝑇 ⊆ Word (Base‘𝐺)) |
| 24 | | psgnunilem2.w |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ Word 𝑇) |
| 25 | 23, 24 | sseldd 3984 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺)) |
| 26 | | pfxcl 14715 |
. . . . . . . . 9
⊢ (𝑊 ∈ Word (Base‘𝐺) → (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺)) |
| 27 | 25, 26 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺)) |
| 28 | 20 | gsumwcl 18852 |
. . . . . . . 8
⊢ ((𝐺 ∈ Mnd ∧ (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺)) → (𝐺 Σg (𝑊 prefix 𝐼)) ∈ (Base‘𝐺)) |
| 29 | 18, 27, 28 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (𝐺 Σg (𝑊 prefix 𝐼)) ∈ (Base‘𝐺)) |
| 30 | 15, 20 | symgbasf1o 19392 |
. . . . . . 7
⊢ ((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ (Base‘𝐺) → (𝐺 Σg (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷) |
| 31 | 29, 30 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝐺 Σg (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷) |
| 32 | 31 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷) |
| 33 | | wrdf 14557 |
. . . . . . . . . 10
⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
| 34 | 24, 33 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑇) |
| 35 | | psgnunilem2.ix |
. . . . . . . . . 10
⊢ (𝜑 → 𝐼 ∈ (0..^𝐿)) |
| 36 | | psgnunilem2.l |
. . . . . . . . . . 11
⊢ (𝜑 → (♯‘𝑊) = 𝐿) |
| 37 | 36 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿)) |
| 38 | 35, 37 | eleqtrrd 2844 |
. . . . . . . . 9
⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊))) |
| 39 | 34, 38 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇) |
| 40 | 21, 39 | sselid 3981 |
. . . . . . 7
⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺)) |
| 41 | 15, 20 | symgbasf1o 19392 |
. . . . . . 7
⊢ ((𝑊‘𝐼) ∈ (Base‘𝐺) → (𝑊‘𝐼):𝐷–1-1-onto→𝐷) |
| 42 | 40, 41 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑊‘𝐼):𝐷–1-1-onto→𝐷) |
| 43 | 42 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊‘𝐼):𝐷–1-1-onto→𝐷) |
| 44 | 15, 20 | symgsssg 19485 |
. . . . . . . . . . 11
⊢ (𝐷 ∈ 𝑉 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubGrp‘𝐺)) |
| 45 | | subgsubm 19166 |
. . . . . . . . . . 11
⊢ ({𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} ∈
(SubGrp‘𝐺) →
{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} ∈
(SubMnd‘𝐺)) |
| 46 | 14, 44, 45 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺)) |
| 47 | | fzossfz 13718 |
. . . . . . . . . . . . . . . 16
⊢
(0..^𝐿) ⊆
(0...𝐿) |
| 48 | 47, 35 | sselid 3981 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐼 ∈ (0...𝐿)) |
| 49 | 36 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (0...(♯‘𝑊)) = (0...𝐿)) |
| 50 | 48, 49 | eleqtrrd 2844 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐼 ∈ (0...(♯‘𝑊))) |
| 51 | | pfxmpt 14716 |
. . . . . . . . . . . . . 14
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝐼 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐼) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠))) |
| 52 | 24, 50, 51 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑊 prefix 𝐼) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠))) |
| 53 | | difeq1 4119 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑊‘𝑠) → (𝑗 ∖ I ) = ((𝑊‘𝑠) ∖ I )) |
| 54 | 53 | dmeqd 5916 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑊‘𝑠) → dom (𝑗 ∖ I ) = dom ((𝑊‘𝑠) ∖ I )) |
| 55 | 54 | sseq1d 4015 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴}))) |
| 56 | | disj2 4458 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴})) |
| 57 | | disjsn 4711 |
. . . . . . . . . . . . . . . 16
⊢ ((dom
((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
| 58 | 56, 57 | bitr3i 277 |
. . . . . . . . . . . . . . 15
⊢ (dom
((𝑊‘𝑠) ∖ I ) ⊆ (V ∖
{𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
| 59 | 55, 58 | bitrdi 287 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
| 60 | | elfzuz3 13561 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ (ℤ≥‘𝐼)) |
| 61 | 48, 60 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐿 ∈ (ℤ≥‘𝐼)) |
| 62 | 36, 61 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (♯‘𝑊) ∈
(ℤ≥‘𝐼)) |
| 63 | | fzoss2 13727 |
. . . . . . . . . . . . . . . . 17
⊢
((♯‘𝑊)
∈ (ℤ≥‘𝐼) → (0..^𝐼) ⊆ (0..^(♯‘𝑊))) |
| 64 | 62, 63 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0..^𝐼) ⊆ (0..^(♯‘𝑊))) |
| 65 | 64 | sselda 3983 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → 𝑠 ∈ (0..^(♯‘𝑊))) |
| 66 | 34 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑠) ∈ 𝑇) |
| 67 | 21, 66 | sselid 3981 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑠) ∈ (Base‘𝐺)) |
| 68 | 65, 67 | syldan 591 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ (Base‘𝐺)) |
| 69 | | psgnunilem2.al |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I )) |
| 70 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 = 𝑠 → (𝑊‘𝑘) = (𝑊‘𝑠)) |
| 71 | 70 | difeq1d 4125 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝑠 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑠) ∖ I )) |
| 72 | 71 | dmeqd 5916 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝑠 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑠) ∖ I )) |
| 73 | 72 | eleq2d 2827 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝑠 → (𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
| 74 | 73 | notbid 318 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝑠 → (¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))) |
| 75 | 74 | cbvralvw 3237 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑘 ∈
(0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
| 76 | 69, 75 | sylib 218 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
| 77 | 76 | r19.21bi 3251 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )) |
| 78 | 59, 68, 77 | elrabd 3694 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
| 79 | 52, 78 | fmpt3d 7136 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑊 prefix 𝐼):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
| 80 | 79 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 prefix 𝐼):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
| 81 | | iswrdi 14556 |
. . . . . . . . . . 11
⊢ ((𝑊 prefix 𝐼):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → (𝑊 prefix 𝐼) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
| 82 | 80, 81 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 prefix 𝐼) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
| 83 | | gsumwsubmcl 18850 |
. . . . . . . . . 10
⊢ (({𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖
{𝐴})} ∈
(SubMnd‘𝐺) ∧
(𝑊 prefix 𝐼) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) → (𝐺 Σg (𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
| 84 | 46, 82, 83 | syl2an2r 685 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) |
| 85 | | difeq1 4119 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝐺 Σg (𝑊 prefix 𝐼)) → (𝑗 ∖ I ) = ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
| 86 | 85 | dmeqd 5916 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝐺 Σg (𝑊 prefix 𝐼)) → dom (𝑗 ∖ I ) = dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
| 87 | 86 | sseq1d 4015 |
. . . . . . . . . . . 12
⊢ (𝑗 = (𝐺 Σg (𝑊 prefix 𝐼)) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝐺 Σg
(𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴}))) |
| 88 | 87 | elrab 3692 |
. . . . . . . . . . 11
⊢ ((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ↔ ((𝐺 Σg (𝑊 prefix 𝐼)) ∈ (Base‘𝐺) ∧ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴}))) |
| 89 | 88 | simprbi 496 |
. . . . . . . . . 10
⊢ ((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → dom ((𝐺 Σg
(𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴})) |
| 90 | | disj2 4458 |
. . . . . . . . . . 11
⊢ ((dom
((𝐺
Σg (𝑊 prefix 𝐼)) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴})) |
| 91 | | disjsn 4711 |
. . . . . . . . . . 11
⊢ ((dom
((𝐺
Σg (𝑊 prefix 𝐼)) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
| 92 | 90, 91 | bitr3i 277 |
. . . . . . . . . 10
⊢ (dom
((𝐺
Σg (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
| 93 | 89, 92 | sylib 218 |
. . . . . . . . 9
⊢ ((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
| 94 | 84, 93 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I )) |
| 95 | | psgnunilem2.a |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) |
| 96 | 95 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) |
| 97 | 94, 96 | jca 511 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) |
| 98 | 97 | olcd 875 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))) |
| 99 | | excxor 1516 |
. . . . . 6
⊢ ((𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ↔ ((𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))) |
| 100 | 98, 99 | sylibr 234 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) |
| 101 | | f1omvdco3 19467 |
. . . . 5
⊢ (((𝐺 Σg
(𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷 ∧ (𝑊‘𝐼):𝐷–1-1-onto→𝐷 ∧ (𝐴 ∈ dom ((𝐺 Σg (𝑊 prefix 𝐼)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) → 𝐴 ∈ dom (((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I )) |
| 102 | 32, 43, 100, 101 | syl3anc 1373 |
. . . 4
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom (((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I )) |
| 103 | | elfzo0 13740 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ (0..^𝐿) ↔ (𝐼 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ∧ 𝐼 < 𝐿)) |
| 104 | 103 | simp2bi 1147 |
. . . . . . . . . . . . . 14
⊢ (𝐼 ∈ (0..^𝐿) → 𝐿 ∈ ℕ) |
| 105 | 35, 104 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐿 ∈ ℕ) |
| 106 | 36, 105 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢ (𝜑 → (♯‘𝑊) ∈
ℕ) |
| 107 | | wrdfin 14570 |
. . . . . . . . . . . . 13
⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin) |
| 108 | | hashnncl 14405 |
. . . . . . . . . . . . 13
⊢ (𝑊 ∈ Fin →
((♯‘𝑊) ∈
ℕ ↔ 𝑊 ≠
∅)) |
| 109 | 24, 107, 108 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((♯‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅)) |
| 110 | 106, 109 | mpbid 232 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ≠ ∅) |
| 111 | 110 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 ≠ ∅) |
| 112 | | pfxlswccat 14751 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉) = 𝑊) |
| 113 | 112 | eqcomd 2743 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → 𝑊 = ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉)) |
| 114 | 24, 111, 113 | syl2an2r 685 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉)) |
| 115 | 36 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((♯‘𝑊) − 1) = (𝐿 − 1)) |
| 116 | 115 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((♯‘𝑊) − 1) = (𝐿 − 1)) |
| 117 | 105 | nncnd 12282 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐿 ∈ ℂ) |
| 118 | | 1cnd 11256 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ∈
ℂ) |
| 119 | | elfzoelz 13699 |
. . . . . . . . . . . . . . 15
⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℤ) |
| 120 | 35, 119 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐼 ∈ ℤ) |
| 121 | 120 | zcnd 12723 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐼 ∈ ℂ) |
| 122 | 117, 118,
121 | subadd2d 11639 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐿 − 1) = 𝐼 ↔ (𝐼 + 1) = 𝐿)) |
| 123 | 122 | biimpar 477 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐿 − 1) = 𝐼) |
| 124 | 116, 123 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((♯‘𝑊) − 1) = 𝐼) |
| 125 | | oveq2 7439 |
. . . . . . . . . . . 12
⊢
(((♯‘𝑊)
− 1) = 𝐼 →
(𝑊 prefix
((♯‘𝑊) −
1)) = (𝑊 prefix 𝐼)) |
| 126 | 125 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → (𝑊 prefix ((♯‘𝑊) − 1)) = (𝑊 prefix 𝐼)) |
| 127 | | lsw 14602 |
. . . . . . . . . . . . . 14
⊢ (𝑊 ∈ Word 𝑇 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
| 128 | 24, 127 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1))) |
| 129 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢
(((♯‘𝑊)
− 1) = 𝐼 →
(𝑊‘((♯‘𝑊) − 1)) = (𝑊‘𝐼)) |
| 130 | 128, 129 | sylan9eq 2797 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → (lastS‘𝑊) = (𝑊‘𝐼)) |
| 131 | 130 | s1eqd 14639 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → 〈“(lastS‘𝑊)”〉 =
〈“(𝑊‘𝐼)”〉) |
| 132 | 126, 131 | oveq12d 7449 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉) = ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) |
| 133 | 124, 132 | syldan 591 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++
〈“(lastS‘𝑊)”〉) = ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) |
| 134 | 114, 133 | eqtrd 2777 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) |
| 135 | 134 | oveq2d 7447 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = (𝐺 Σg ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉))) |
| 136 | 40 | s1cld 14641 |
. . . . . . . . 9
⊢ (𝜑 → 〈“(𝑊‘𝐼)”〉 ∈ Word (Base‘𝐺)) |
| 137 | | eqid 2737 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 138 | 20, 137 | gsumccat 18854 |
. . . . . . . . 9
⊢ ((𝐺 ∈ Mnd ∧ (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺) ∧ 〈“(𝑊‘𝐼)”〉 ∈ Word (Base‘𝐺)) → (𝐺 Σg ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉))) |
| 139 | 18, 27, 136, 138 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → (𝐺 Σg ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉))) |
| 140 | 139 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg ((𝑊 prefix 𝐼) ++ 〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉))) |
| 141 | 20 | gsumws1 18851 |
. . . . . . . . . . 11
⊢ ((𝑊‘𝐼) ∈ (Base‘𝐺) → (𝐺 Σg
〈“(𝑊‘𝐼)”〉) = (𝑊‘𝐼)) |
| 142 | 40, 141 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg
〈“(𝑊‘𝐼)”〉) = (𝑊‘𝐼)) |
| 143 | 142 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg
(𝑊 prefix 𝐼))(+g‘𝐺)(𝑊‘𝐼))) |
| 144 | 15, 20, 137 | symgov 19401 |
. . . . . . . . . 10
⊢ (((𝐺 Σg
(𝑊 prefix 𝐼)) ∈ (Base‘𝐺) ∧ (𝑊‘𝐼) ∈ (Base‘𝐺)) → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
| 145 | 29, 40, 144 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
| 146 | 143, 145 | eqtrd 2777 |
. . . . . . . 8
⊢ (𝜑 → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg
(𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
| 147 | 146 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σg (𝑊 prefix 𝐼))(+g‘𝐺)(𝐺 Σg
〈“(𝑊‘𝐼)”〉)) = ((𝐺 Σg
(𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
| 148 | 135, 140,
147 | 3eqtrd 2781 |
. . . . . 6
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = ((𝐺 Σg (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼))) |
| 149 | 148 | difeq1d 4125 |
. . . . 5
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σg 𝑊) ∖ I ) = (((𝐺 Σg
(𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I )) |
| 150 | 149 | dmeqd 5916 |
. . . 4
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (((𝐺 Σg
(𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I )) |
| 151 | 102, 150 | eleqtrrd 2844 |
. . 3
⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I )) |
| 152 | 13, 151 | mtand 816 |
. 2
⊢ (𝜑 → ¬ (𝐼 + 1) = 𝐿) |
| 153 | | fzostep1 13822 |
. . . 4
⊢ (𝐼 ∈ (0..^𝐿) → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿)) |
| 154 | 35, 153 | syl 17 |
. . 3
⊢ (𝜑 → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿)) |
| 155 | 154 | ord 865 |
. 2
⊢ (𝜑 → (¬ (𝐼 + 1) ∈ (0..^𝐿) → (𝐼 + 1) = 𝐿)) |
| 156 | 152, 155 | mt3d 148 |
1
⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿)) |