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Theorem setsstruct 16729
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 16705 . . . . . 6 (𝐺 Struct ⟨𝑀, 𝑁⟩ ↔ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ Fun (𝐺 ∖ {∅}) ∧ dom 𝐺 ⊆ (𝑀...𝑁)))
2 simp2 1139 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝐺 Struct ⟨𝑀, 𝑁⟩)
3 simp3l 1203 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝐸𝑉)
4 1z 12207 . . . . . . . . . . . . . . . 16 1 ∈ ℤ
5 nnge1 11858 . . . . . . . . . . . . . . . 16 (𝑀 ∈ ℕ → 1 ≤ 𝑀)
6 eluzuzle 12447 . . . . . . . . . . . . . . . 16 ((1 ∈ ℤ ∧ 1 ≤ 𝑀) → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ (ℤ‘1)))
74, 5, 6sylancr 590 . . . . . . . . . . . . . . 15 (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ (ℤ‘1)))
8 elnnuz 12478 . . . . . . . . . . . . . . 15 (𝐼 ∈ ℕ ↔ 𝐼 ∈ (ℤ‘1))
97, 8syl6ibr 255 . . . . . . . . . . . . . 14 (𝑀 ∈ ℕ → (𝐼 ∈ (ℤ𝑀) → 𝐼 ∈ ℕ))
109adantld 494 . . . . . . . . . . . . 13 (𝑀 ∈ ℕ → ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → 𝐼 ∈ ℕ))
11103ad2ant1 1135 . . . . . . . . . . . 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 1113 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝐼 ∈ ℕ)
142, 3, 133jca 1130 . . . . . . . . 9 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → (𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ 𝐸𝑉𝐼 ∈ ℕ))
15 op1stg 7773 . . . . . . . . . . . . . . . 16 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (1st ‘⟨𝑀, 𝑁⟩) = 𝑀)
1615breq2d 5065 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩) ↔ 𝐼𝑀))
17 eqidd 2738 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐼 = 𝐼)
1816, 17, 15ifbieq12d 4467 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
19183adant3 1134 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
2019adantr 484 . . . . . . . . . . . 12 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)) = if(𝐼𝑀, 𝐼, 𝑀))
21 eluz2 12444 . . . . . . . . . . . . . . . 16 (𝐼 ∈ (ℤ𝑀) ↔ (𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼))
22 zre 12180 . . . . . . . . . . . . . . . . . . . 20 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ)
2322rexrd 10883 . . . . . . . . . . . . . . . . . . 19 (𝐼 ∈ ℤ → 𝐼 ∈ ℝ*)
24233ad2ant2 1136 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝐼 ∈ ℝ*)
25 zre 12180 . . . . . . . . . . . . . . . . . . . 20 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
2625rexrd 10883 . . . . . . . . . . . . . . . . . . 19 (𝑀 ∈ ℤ → 𝑀 ∈ ℝ*)
27263ad2ant1 1135 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝑀 ∈ ℝ*)
28 simp3 1140 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℤ ∧ 𝐼 ∈ ℤ ∧ 𝑀𝐼) → 𝑀𝐼)
2924, 27, 283jca 1130 . . . . . . . . . . . . . . . . 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 220 . . . . . . . . . . . . . . 15 (𝐼 ∈ (ℤ𝑀) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼)))
3231adantl 485 . . . . . . . . . . . . . 14 ((𝐸𝑉𝐼 ∈ (ℤ𝑀)) → ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼)))
3332impcom 411 . . . . . . . . . . . . 13 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → (𝐼 ∈ ℝ*𝑀 ∈ ℝ*𝑀𝐼))
34 xrmineq 12770 . . . . . . . . . . . . 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 2778 . . . . . . . . . . 11 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝑀 = if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)))
37363adant2 1133 . . . . . . . . . 10 (((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) ∧ 𝐺 Struct ⟨𝑀, 𝑁⟩ ∧ (𝐸𝑉𝐼 ∈ (ℤ𝑀))) → 𝑀 = if(𝐼 ≤ (1st ‘⟨𝑀, 𝑁⟩), 𝐼, (1st ‘⟨𝑀, 𝑁⟩)))
38 op2ndg 7774 . . . . . . . . . . . . . . 15 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (2nd ‘⟨𝑀, 𝑁⟩) = 𝑁)
3938eqcomd 2743 . . . . . . . . . . . . . 14 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 = (2nd ‘⟨𝑀, 𝑁⟩))
4039breq2d 5065 . . . . . . . . . . . . 13 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐼𝑁𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩)))
4140, 39, 17ifbieq12d 4467 . . . . . . . . . . . 12 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → if(𝐼𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
42413adant3 1134 . . . . . . . . . . 11 ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑀𝑁) → if(𝐼𝑁, 𝑁, 𝐼) = if(𝐼 ≤ (2nd ‘⟨𝑀, 𝑁⟩), (2nd ‘⟨𝑀, 𝑁⟩), 𝐼))
43423ad2ant1 1135 . . . . . . . . . 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 4792 . . . . . . . . 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 515 . . . . . . . 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 1121 . . . . . . 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 1135 . . . . . 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 220 . . . . 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 418 . . 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 1113 . 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 16727 . 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 399  w3a 1089   = wceq 1543  wcel 2110  cdif 3863  wss 3866  c0 4237  ifcif 4439  {csn 4541  cop 4547   class class class wbr 5053  dom cdm 5551  Fun wfun 6374  cfv 6380  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  1c1 10730  *cxr 10866  cle 10868  cn 11830  cz 12176  cuz 12438  ...cfz 13095   Struct cstr 16699   sSet csts 16716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-n0 12091  df-z 12177  df-uz 12439  df-fz 13096  df-struct 16700  df-sets 16717
This theorem is referenced by: (None)
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