| Step | Hyp | Ref
| Expression |
| 1 | | fzssz 13566 |
. . . . . . . . 9
⊢ (𝑀...𝑁) ⊆ ℤ |
| 2 | | zssre 12620 |
. . . . . . . . 9
⊢ ℤ
⊆ ℝ |
| 3 | 1, 2 | sstri 3993 |
. . . . . . . 8
⊢ (𝑀...𝑁) ⊆ ℝ |
| 4 | | ltso 11341 |
. . . . . . . 8
⊢ < Or
ℝ |
| 5 | | soss 5612 |
. . . . . . . 8
⊢ ((𝑀...𝑁) ⊆ ℝ → ( < Or ℝ
→ < Or (𝑀...𝑁))) |
| 6 | 3, 4, 5 | mp2 9 |
. . . . . . 7
⊢ < Or
(𝑀...𝑁) |
| 7 | | fzfi 14013 |
. . . . . . 7
⊢ (𝑀...𝑁) ∈ Fin |
| 8 | | fz1iso 14501 |
. . . . . . 7
⊢ (( <
Or (𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → ∃ℎ ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁))) |
| 9 | 6, 7, 8 | mp2an 692 |
. . . . . 6
⊢
∃ℎ ℎ Isom < , <
((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) |
| 10 | | fzisoeu.4 |
. . . . . . . . . . . . . . . 16
⊢ 𝑁 = ((♯‘𝐻) + (𝑀 − 1)) |
| 11 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐻 = ∅ →
(♯‘𝐻) =
(♯‘∅)) |
| 12 | | hash0 14406 |
. . . . . . . . . . . . . . . . . 18
⊢
(♯‘∅) = 0 |
| 13 | 11, 12 | eqtrdi 2793 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐻 = ∅ →
(♯‘𝐻) =
0) |
| 14 | 13 | oveq1d 7446 |
. . . . . . . . . . . . . . . 16
⊢ (𝐻 = ∅ →
((♯‘𝐻) + (𝑀 − 1)) = (0 + (𝑀 − 1))) |
| 15 | 10, 14 | eqtrid 2789 |
. . . . . . . . . . . . . . 15
⊢ (𝐻 = ∅ → 𝑁 = (0 + (𝑀 − 1))) |
| 16 | 15 | oveq2d 7447 |
. . . . . . . . . . . . . 14
⊢ (𝐻 = ∅ → (𝑀...𝑁) = (𝑀...(0 + (𝑀 − 1)))) |
| 17 | 16 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...𝑁) = (𝑀...(0 + (𝑀 − 1)))) |
| 18 | | fzisoeu.m |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 19 | 18 | zcnd 12723 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑀 ∈ ℂ) |
| 20 | | 1cnd 11256 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 1 ∈
ℂ) |
| 21 | 19, 20 | subcld 11620 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) ∈ ℂ) |
| 22 | 21 | addlidd 11462 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (0 + (𝑀 − 1)) = (𝑀 − 1)) |
| 23 | 22 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = (𝑀...(𝑀 − 1))) |
| 24 | 18 | zred 12722 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ ℝ) |
| 25 | 24 | ltm1d 12200 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑀 − 1) < 𝑀) |
| 26 | | peano2zm 12660 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈
ℤ) |
| 27 | 18, 26 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) ∈ ℤ) |
| 28 | | fzn 13580 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ)
→ ((𝑀 − 1) <
𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) |
| 29 | 18, 27, 28 | syl2anc 584 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅)) |
| 30 | 25, 29 | mpbid 232 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀...(𝑀 − 1)) = ∅) |
| 31 | 23, 30 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀...(0 + (𝑀 − 1))) = ∅) |
| 32 | 31 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...(0 + (𝑀 − 1))) = ∅) |
| 33 | | eqcom 2744 |
. . . . . . . . . . . . . . 15
⊢ (𝐻 = ∅ ↔ ∅ =
𝐻) |
| 34 | 33 | biimpi 216 |
. . . . . . . . . . . . . 14
⊢ (𝐻 = ∅ → ∅ =
𝐻) |
| 35 | 34 | adantl 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝐻 = ∅) → ∅ = 𝐻) |
| 36 | 17, 32, 35 | 3eqtrd 2781 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝐻 = ∅) → (𝑀...𝑁) = 𝐻) |
| 37 | 36 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = (♯‘𝐻)) |
| 38 | 20, 19 | pncan3d 11623 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (1 + (𝑀 − 1)) = 𝑀) |
| 39 | 38 | eqcomd 2743 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 = (1 + (𝑀 − 1))) |
| 40 | 39 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 = (1 + (𝑀 − 1))) |
| 41 | | 1red 11262 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ∈
ℝ) |
| 42 | | neqne 2948 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (¬
𝐻 = ∅ → 𝐻 ≠ ∅) |
| 43 | 42 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝐻 ≠ ∅) |
| 44 | | fzisoeu.h |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐻 ∈ Fin) |
| 45 | 44 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝐻 ∈ Fin) |
| 46 | | hashnncl 14405 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐻 ∈ Fin →
((♯‘𝐻) ∈
ℕ ↔ 𝐻 ≠
∅)) |
| 47 | 45, 46 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((♯‘𝐻) ∈ ℕ ↔ 𝐻 ≠ ∅)) |
| 48 | 43, 47 | mpbird 257 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘𝐻) ∈
ℕ) |
| 49 | 48 | nnred 12281 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘𝐻) ∈
ℝ) |
| 50 | 27 | zred 12722 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 − 1) ∈ ℝ) |
| 51 | 50 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑀 − 1) ∈ ℝ) |
| 52 | 48 | nnge1d 12314 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 1 ≤
(♯‘𝐻)) |
| 53 | 41, 49, 51, 52 | leadd1dd 11877 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ ((♯‘𝐻) + (𝑀 − 1))) |
| 54 | 53, 10 | breqtrrdi 5185 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (1 + (𝑀 − 1)) ≤ 𝑁) |
| 55 | 40, 54 | eqbrtrd 5165 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ≤ 𝑁) |
| 56 | 18 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑀 ∈ ℤ) |
| 57 | | hashcl 14395 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐻 ∈ Fin →
(♯‘𝐻) ∈
ℕ0) |
| 58 | | nn0z 12638 |
. . . . . . . . . . . . . . . . . . 19
⊢
((♯‘𝐻)
∈ ℕ0 → (♯‘𝐻) ∈ ℤ) |
| 59 | 44, 57, 58 | 3syl 18 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (♯‘𝐻) ∈
ℤ) |
| 60 | 59, 27 | zaddcld 12726 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((♯‘𝐻) + (𝑀 − 1)) ∈
ℤ) |
| 61 | 10, 60 | eqeltrid 2845 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 62 | 61 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ ℤ) |
| 63 | | eluz 12892 |
. . . . . . . . . . . . . . 15
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈
(ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
| 64 | 56, 62, 63 | syl2anc 584 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁)) |
| 65 | 55, 64 | mpbird 257 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 66 | | hashfz 14466 |
. . . . . . . . . . . . 13
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (♯‘(𝑀...𝑁)) = ((𝑁 − 𝑀) + 1)) |
| 67 | 65, 66 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = ((𝑁 − 𝑀) + 1)) |
| 68 | 10 | oveq1i 7441 |
. . . . . . . . . . . . . . . 16
⊢ (𝑁 − 𝑀) = (((♯‘𝐻) + (𝑀 − 1)) − 𝑀) |
| 69 | 44, 57 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (♯‘𝐻) ∈
ℕ0) |
| 70 | 69 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (♯‘𝐻) ∈
ℂ) |
| 71 | 70, 21, 19 | addsubassd 11640 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (((♯‘𝐻) + (𝑀 − 1)) − 𝑀) = ((♯‘𝐻) + ((𝑀 − 1) − 𝑀))) |
| 72 | 68, 71 | eqtrid 2789 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑁 − 𝑀) = ((♯‘𝐻) + ((𝑀 − 1) − 𝑀))) |
| 73 | 20 | negcld 11607 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → -1 ∈
ℂ) |
| 74 | 19, 20 | negsubd 11626 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 + -1) = (𝑀 − 1)) |
| 75 | 74 | eqcomd 2743 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 − 1) = (𝑀 + -1)) |
| 76 | 19, 73, 75 | mvrladdd 11676 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝑀 − 1) − 𝑀) = -1) |
| 77 | 76 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((♯‘𝐻) + ((𝑀 − 1) − 𝑀)) = ((♯‘𝐻) + -1)) |
| 78 | 72, 77 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑁 − 𝑀) = ((♯‘𝐻) + -1)) |
| 79 | 78 | oveq1d 7446 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑁 − 𝑀) + 1) = (((♯‘𝐻) + -1) + 1)) |
| 80 | 79 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → ((𝑁 − 𝑀) + 1) = (((♯‘𝐻) + -1) + 1)) |
| 81 | 70, 20 | negsubd 11626 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((♯‘𝐻) + -1) = ((♯‘𝐻) − 1)) |
| 82 | 81 | oveq1d 7446 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((♯‘𝐻) + -1) + 1) =
(((♯‘𝐻) −
1) + 1)) |
| 83 | 70, 20 | npcand 11624 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((♯‘𝐻) − 1) + 1) =
(♯‘𝐻)) |
| 84 | 82, 83 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((♯‘𝐻) + -1) + 1) =
(♯‘𝐻)) |
| 85 | 84 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (((♯‘𝐻) + -1) + 1) =
(♯‘𝐻)) |
| 86 | 67, 80, 85 | 3eqtrd 2781 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬ 𝐻 = ∅) → (♯‘(𝑀...𝑁)) = (♯‘𝐻)) |
| 87 | 37, 86 | pm2.61dan 813 |
. . . . . . . . . 10
⊢ (𝜑 → (♯‘(𝑀...𝑁)) = (♯‘𝐻)) |
| 88 | 87 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → (1...(♯‘(𝑀...𝑁))) = (1...(♯‘𝐻))) |
| 89 | | isoeq4 7340 |
. . . . . . . . 9
⊢
((1...(♯‘(𝑀...𝑁))) = (1...(♯‘𝐻)) → (ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) ↔ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)))) |
| 90 | 88, 89 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) ↔ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)))) |
| 91 | 90 | biimpd 229 |
. . . . . . 7
⊢ (𝜑 → (ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) → ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)))) |
| 92 | 91 | eximdv 1917 |
. . . . . 6
⊢ (𝜑 → (∃ℎ ℎ Isom < , < ((1...(♯‘(𝑀...𝑁))), (𝑀...𝑁)) → ∃ℎ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)))) |
| 93 | 9, 92 | mpi 20 |
. . . . 5
⊢ (𝜑 → ∃ℎ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁))) |
| 94 | | fzisoeu.or |
. . . . . 6
⊢ (𝜑 → < Or 𝐻) |
| 95 | | fz1iso 14501 |
. . . . . 6
⊢ (( <
Or 𝐻 ∧ 𝐻 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) |
| 96 | 94, 44, 95 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) |
| 97 | | exdistrv 1955 |
. . . . 5
⊢
(∃ℎ∃𝑔(ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) ↔ (∃ℎ ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) |
| 98 | 93, 96, 97 | sylanbrc 583 |
. . . 4
⊢ (𝜑 → ∃ℎ∃𝑔(ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) |
| 99 | | isocnv 7350 |
. . . . . . . 8
⊢ (ℎ Isom < , <
((1...(♯‘𝐻)),
(𝑀...𝑁)) → ◡ℎ Isom < , < ((𝑀...𝑁), (1...(♯‘𝐻)))) |
| 100 | 99 | ad2antrl 728 |
. . . . . . 7
⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) → ◡ℎ Isom < , < ((𝑀...𝑁), (1...(♯‘𝐻)))) |
| 101 | | simprr 773 |
. . . . . . 7
⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) → 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) |
| 102 | | isotr 7356 |
. . . . . . 7
⊢ ((◡ℎ Isom < , < ((𝑀...𝑁), (1...(♯‘𝐻))) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) → (𝑔 ∘ ◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)) |
| 103 | 100, 101,
102 | syl2anc 584 |
. . . . . 6
⊢ ((𝜑 ∧ (ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻))) → (𝑔 ∘ ◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)) |
| 104 | 103 | ex 412 |
. . . . 5
⊢ (𝜑 → ((ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) → (𝑔 ∘ ◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))) |
| 105 | 104 | 2eximdv 1919 |
. . . 4
⊢ (𝜑 → (∃ℎ∃𝑔(ℎ Isom < , < ((1...(♯‘𝐻)), (𝑀...𝑁)) ∧ 𝑔 Isom < , < ((1...(♯‘𝐻)), 𝐻)) → ∃ℎ∃𝑔(𝑔 ∘ ◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))) |
| 106 | 98, 105 | mpd 15 |
. . 3
⊢ (𝜑 → ∃ℎ∃𝑔(𝑔 ∘ ◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻)) |
| 107 | | vex 3484 |
. . . . . . 7
⊢ 𝑔 ∈ V |
| 108 | | vex 3484 |
. . . . . . . 8
⊢ ℎ ∈ V |
| 109 | 108 | cnvex 7947 |
. . . . . . 7
⊢ ◡ℎ ∈ V |
| 110 | 107, 109 | coex 7952 |
. . . . . 6
⊢ (𝑔 ∘ ◡ℎ) ∈ V |
| 111 | | isoeq1 7337 |
. . . . . 6
⊢ (𝑓 = (𝑔 ∘ ◡ℎ) → (𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (𝑔 ∘ ◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻))) |
| 112 | 110, 111 | spcev 3606 |
. . . . 5
⊢ ((𝑔 ∘ ◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻) → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)) |
| 113 | 112 | a1i 11 |
. . . 4
⊢ (𝜑 → ((𝑔 ∘ ◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻) → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))) |
| 114 | 113 | exlimdvv 1934 |
. . 3
⊢ (𝜑 → (∃ℎ∃𝑔(𝑔 ∘ ◡ℎ) Isom < , < ((𝑀...𝑁), 𝐻) → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))) |
| 115 | 106, 114 | mpd 15 |
. 2
⊢ (𝜑 → ∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)) |
| 116 | | ltwefz 14004 |
. . 3
⊢ < We
(𝑀...𝑁) |
| 117 | | wemoiso 7998 |
. . 3
⊢ ( < We
(𝑀...𝑁) → ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)) |
| 118 | 116, 117 | mp1i 13 |
. 2
⊢ (𝜑 → ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)) |
| 119 | | df-eu 2569 |
. 2
⊢
(∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ↔ (∃𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻) ∧ ∃*𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻))) |
| 120 | 115, 118,
119 | sylanbrc 583 |
1
⊢ (𝜑 → ∃!𝑓 𝑓 Isom < , < ((𝑀...𝑁), 𝐻)) |