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

Theorem setsstruct 15819
Description: An extensible structure with a replaced slot is an extensible structure. (Contributed by AV, 9-Jun-2021.) (Revised by AV, 14-Nov-2021.)
Assertion
Ref Expression
setsstruct ((𝐸𝑉𝐼 ∈ (ℤ𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩)

Proof of Theorem setsstruct
StepHypRef Expression
1 isstruct 15793 . . . . . 6 (𝐺 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)))
2 simp2 1060 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝐺 Struct ⟨𝑀, 𝑁⟩)
3 simp3l 1087 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝐸𝑉)
4 1z 11351 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
5 nnge1 10990 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ → 1 ≤ 𝑀)
6 eluzuzle 11640 . . . . . . . . . . . . . . . 16 ((1 ∈ ℤ ∧ 1 ≤ 𝑀) → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ (ℤ‘1)))
74, 5, 6sylancr 694 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ (ℤ‘1)))
8 elnnuz 11668 . . . . . . . . . . . . . . 15 (𝐼 ∈ ℕ ↔ 𝐼 ∈ (ℤ‘1))
97, 8syl6ibr 242 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ ℕ))
109adantld 483 . . . . . . . . . . . . 13 (𝑀 ∈ ℕ → ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → 𝐼 ∈ ℕ))
11103ad2ant1 1080 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → 𝐼 ∈ ℕ))
1211a1d 25 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → 𝐼 ∈ ℕ)))
13123imp 1254 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝐼 ∈ ℕ)
142, 3, 133jca 1240 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → (𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ))
15 op1stg 7125 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
1615breq2d 4625 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩) ↔ 𝐼𝑀))
17 eqidd 2622 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐼 = 𝐼)
1816, 17, 15ifbieq12d 4085 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
19183adant3 1079 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
2019adantr 481 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
21 eluz2 11637 . . . . . . . . . . . . . . . 16 (𝐼 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼))
22 zre 11325 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
2322rexrd 10033 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*)
24233ad2ant2 1081 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝐼 ∈ ℝ*)
25 zre 11325 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
2625rexrd 10033 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
27263ad2ant1 1080 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝑀 ∈ ℝ*)
28 simp3 1061 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝑀𝐼)
2924, 27, 283jca 1240 . . . . . . . . . . . . . . . . 17 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼))
3029a1d 25 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼)))
3121, 30sylbi 207 . . . . . . . . . . . . . . 15 (𝐼 ∈ (ℤ𝑀) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼)))
3231adantl 482 . . . . . . . . . . . . . 14 ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼)))
3332impcom 446 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼))
34 xrmineq 11954 . . . . . . . . . . . . 13 ((𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼) → if(𝐼𝑀, 𝐼, 𝑀) = 𝑀)
3533, 34syl 17 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → if(𝐼𝑀, 𝐼, 𝑀) = 𝑀)
3620, 35eqtr2d 2656 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝑀 = if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)))
37363adant2 1078 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝑀 = if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)))
38 op2ndg 7126 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3938eqcomd 2627 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 = (2nd ‘⟨𝑀, 𝑁⟩))
4039breq2d 4625 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼𝑁𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩)))
4140, 39, 17ifbieq12d 4085 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
42413adant3 1079 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → if(𝐼𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
43423ad2ant1 1080 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → if(𝐼𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
4437, 43opeq12d 4378 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩)
4514, 44jca 554 . . . . . . . 8 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))
46453exp 1261 . . . . . . 7 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))))
47463ad2ant1 1080 . . . . . 6 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))))
481, 47sylbi 207 . . . . 5 (𝐺 Struct ⟨𝑀, 𝑁⟩ → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))))
4948pm2.43i 52 . . . 4 (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩)))
5049expdcom 455 . . 3 (𝐸𝑉 → (𝐼 ∈ (ℤ𝑀) → (𝐺 Struct ⟨𝑀, 𝑁⟩ → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))))
51503imp 1254 . 2 ((𝐸𝑉𝐼 ∈ (ℤ𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → ((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩))
52 setsstruct2 15817 . 2 (((𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ) ∧ ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩ = ⟨if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)), if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼)⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩)
5351, 52syl 17 1 ((𝐸𝑉𝐼 ∈ (ℤ𝑀) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩) → (𝐺 sSet ⟨𝐼, 𝐸⟩) Struct ⟨𝑀, if(𝐼𝑁, 𝑁, 𝐼)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cdif 3552  wss 3555  c0 3891  ifcif 4058  {csn 4148  cop 4154   class class class wbr 4613  dom cdm 5074  Fun wfun 5841  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  1c1 9881  *cxr 10017  cle 10019  cn 10964  cz 11321  cuz 11631  ...cfz 12268   Struct cstr 15777   sSet csts 15779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-struct 15783  df-sets 15787
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator