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Theorem setsstruct 17048
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 17024 . . . . . 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 12533 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
5 nnge1 12181 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ → 1 ≤ 𝑀)
6 eluzuzle 12772 . . . . . . . . . . . . . . . 16 ((1 ∈ ℤ ∧ 1 ≤ 𝑀) → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ (ℤ‘1)))
74, 5, 6sylancr 587 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ (ℤ‘1)))
8 elnnuz 12807 . . . . . . . . . . . . . . 15 (𝐼 ∈ ℕ ↔ 𝐼 ∈ (ℤ‘1))
97, 8syl6ibr 251 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ ℕ))
109adantld 491 . . . . . . . . . . . . 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 7933 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
1615breq2d 5117 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩) ↔ 𝐼𝑀))
17 eqidd 2737 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐼 = 𝐼)
1816, 17, 15ifbieq12d 4514 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
21 eluz2 12769 . . . . . . . . . . . . . . . 16 (𝐼 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼))
22 zre 12503 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
2322rexrd 11205 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*)
24233ad2ant2 1134 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝐼 ∈ ℝ*)
25 zre 12503 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
2625rexrd 11205 . . . . . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼))
34 xrmineq 13099 . . . . . . . . . . . . 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 2777 . . . . . . . . . . 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 7934 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3938eqcomd 2742 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 = (2nd ‘⟨𝑀, 𝑁⟩))
4039breq2d 5117 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼𝑁𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩)))
4140, 39, 17ifbieq12d 4514 . . . . . . . . . . . 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 4838 . . . . . . . . 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 512 . . . . . . . 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 216 . . . . 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 415 . . 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 17046 . 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 396  w3a 1087   = wceq 1541  wcel 2106  cdif 3907  wss 3910  c0 4282  ifcif 4486  {csn 4586  cop 4592   class class class wbr 5105  dom cdm 5633  Fun wfun 6490  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  1c1 11052  *cxr 11188  cle 11190  cn 12153  cz 12499  cuz 12763  ...cfz 13424   Struct cstr 17018   sSet csts 17035
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-sets 17036
This theorem is referenced by: (None)
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