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Theorem setsstruct 17225
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 17201 . . . . . 6 (𝐺 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)))
2 simp2 1137 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝐺 Struct ⟨𝑀, 𝑁⟩)
3 simp3l 1201 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝐸𝑉)
4 1z 12675 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
5 nnge1 12323 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ → 1 ≤ 𝑀)
6 eluzuzle 12914 . . . . . . . . . . . . . . . 16 ((1 ∈ ℤ ∧ 1 ≤ 𝑀) → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ (ℤ‘1)))
74, 5, 6sylancr 586 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ (ℤ‘1)))
8 elnnuz 12949 . . . . . . . . . . . . . . 15 (𝐼 ∈ ℕ ↔ 𝐼 ∈ (ℤ‘1))
97, 8imbitrrdi 252 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ ℕ))
109adantld 490 . . . . . . . . . . . . 13 (𝑀 ∈ ℕ → ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → 𝐼 ∈ ℕ))
11103ad2ant1 1133 . . . . . . . . . . . 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 1111 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝐼 ∈ ℕ)
142, 3, 133jca 1128 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → (𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ))
15 op1stg 8044 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
1615breq2d 5178 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩) ↔ 𝐼𝑀))
17 eqidd 2741 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐼 = 𝐼)
1816, 17, 15ifbieq12d 4576 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
19183adant3 1132 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
2019adantr 480 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
21 eluz2 12911 . . . . . . . . . . . . . . . 16 (𝐼 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼))
22 zre 12645 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
2322rexrd 11342 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*)
24233ad2ant2 1134 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝐼 ∈ ℝ*)
25 zre 12645 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
2625rexrd 11342 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
27263ad2ant1 1133 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝑀 ∈ ℝ*)
28 simp3 1138 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝑀𝐼)
2924, 27, 283jca 1128 . . . . . . . . . . . . . . . . 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 217 . . . . . . . . . . . . . . 15 (𝐼 ∈ (ℤ𝑀) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼)))
3231adantl 481 . . . . . . . . . . . . . 14 ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼)))
3332impcom 407 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼))
34 xrmineq 13244 . . . . . . . . . . . . 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 2781 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝑀 = if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)))
37363adant2 1131 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝑀 = if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)))
38 op2ndg 8045 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3938eqcomd 2746 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 = (2nd ‘⟨𝑀, 𝑁⟩))
4039breq2d 5178 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼𝑁𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩)))
4140, 39, 17ifbieq12d 4576 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
42413adant3 1132 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → if(𝐼𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
43423ad2ant1 1133 . . . . . . . . . 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 4905 . . . . . . . . 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 511 . . . . . . . 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 1119 . . . . . . 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 1133 . . . . . 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 217 . . . . 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 414 . . 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 1111 . 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 17223 . 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 395  w3a 1087   = wceq 1537  wcel 2108  cdif 3973  wss 3976  c0 4352  ifcif 4548  {csn 4648  cop 4654   class class class wbr 5166  dom cdm 5700  Fun wfun 6569  cfv 6575  (class class class)co 7450  1st c1st 8030  2nd c2nd 8031  1c1 11187  *cxr 11325  cle 11327  cn 12295  cz 12641  cuz 12905  ...cfz 13569   Struct cstr 17195   sSet csts 17212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7772  ax-cnex 11242  ax-resscn 11243  ax-1cn 11244  ax-icn 11245  ax-addcl 11246  ax-addrcl 11247  ax-mulcl 11248  ax-mulrcl 11249  ax-mulcom 11250  ax-addass 11251  ax-mulass 11252  ax-distr 11253  ax-i2m1 11254  ax-1ne0 11255  ax-1rid 11256  ax-rnegex 11257  ax-rrecex 11258  ax-cnre 11259  ax-pre-lttri 11260  ax-pre-lttrn 11261  ax-pre-ltadd 11262  ax-pre-mulgt0 11263
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6334  df-ord 6400  df-on 6401  df-lim 6402  df-suc 6403  df-iota 6527  df-fun 6577  df-fn 6578  df-f 6579  df-f1 6580  df-fo 6581  df-f1o 6582  df-fv 6583  df-riota 7406  df-ov 7453  df-oprab 7454  df-mpo 7455  df-om 7906  df-1st 8032  df-2nd 8033  df-frecs 8324  df-wrecs 8355  df-recs 8429  df-rdg 8468  df-1o 8524  df-er 8765  df-en 9006  df-dom 9007  df-sdom 9008  df-fin 9009  df-pnf 11328  df-mnf 11329  df-xr 11330  df-ltxr 11331  df-le 11332  df-sub 11524  df-neg 11525  df-nn 12296  df-n0 12556  df-z 12642  df-uz 12906  df-fz 13570  df-struct 17196  df-sets 17213
This theorem is referenced by: (None)
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