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Theorem metakunt24 39364
Description: Technical condition such that metakunt17 39357 holds (Contributed by metakunt, 28-May-2024.)
Hypotheses
Ref Expression
metakunt24.1 (𝜑𝑀 ∈ ℕ)
metakunt24.2 (𝜑𝐼 ∈ ℕ)
metakunt24.3 (𝜑𝐼𝑀)
Assertion
Ref Expression
metakunt24 (𝜑 → ((((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅ ∧ (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}) ∧ (1...𝑀) = (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀})))

Proof of Theorem metakunt24
StepHypRef Expression
1 indir 4205 . . . 4 (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀}))
21a1i 11 . . 3 (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})))
3 metakunt24.1 . . . . . . . 8 (𝜑𝑀 ∈ ℕ)
4 metakunt24.2 . . . . . . . 8 (𝜑𝐼 ∈ ℕ)
5 metakunt24.3 . . . . . . . 8 (𝜑𝐼𝑀)
63, 4, 5metakunt18 39358 . . . . . . 7 (𝜑 → ((((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅) ∧ (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∩ (1...(𝑀𝐼))) = ∅ ∧ ((((𝑀𝐼) + 1)...(𝑀 − 1)) ∩ {𝑀}) = ∅ ∧ ((1...(𝑀𝐼)) ∩ {𝑀}) = ∅)))
76simpld 498 . . . . . 6 (𝜑 → (((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅))
87simp2d 1140 . . . . 5 (𝜑 → ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅)
97simp3d 1141 . . . . 5 (𝜑 → ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅)
108, 9uneq12d 4094 . . . 4 (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = (∅ ∪ ∅))
11 unidm 4082 . . . . 5 (∅ ∪ ∅) = ∅
1211a1i 11 . . . 4 (𝜑 → (∅ ∪ ∅) = ∅)
1310, 12eqtrd 2836 . . 3 (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = ∅)
142, 13eqtrd 2836 . 2 (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
15 1zzd 12005 . . . . 5 (𝜑 → 1 ∈ ℤ)
163nnzd 12078 . . . . 5 (𝜑𝑀 ∈ ℤ)
173nnge1d 11677 . . . . 5 (𝜑 → 1 ≤ 𝑀)
183nnred 11644 . . . . . 6 (𝜑𝑀 ∈ ℝ)
1918leidd 11199 . . . . 5 (𝜑𝑀𝑀)
2015, 16, 16, 17, 19elfzd 12897 . . . 4 (𝜑𝑀 ∈ (1...𝑀))
2120fzsplitnd 39263 . . 3 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
22 oveq1 7146 . . . . . . . . . 10 (𝐼 = 𝑀 → (𝐼 − 1) = (𝑀 − 1))
2322oveq2d 7155 . . . . . . . . 9 (𝐼 = 𝑀 → (1...(𝐼 − 1)) = (1...(𝑀 − 1)))
24 oveq1 7146 . . . . . . . . 9 (𝐼 = 𝑀 → (𝐼...(𝑀 − 1)) = (𝑀...(𝑀 − 1)))
2523, 24uneq12d 4094 . . . . . . . 8 (𝐼 = 𝑀 → ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))))
2625uneq1d 4092 . . . . . . 7 (𝐼 = 𝑀 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
2726adantl 485 . . . . . 6 ((𝜑𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
2818ltm1d 11565 . . . . . . . . . . 11 (𝜑 → (𝑀 − 1) < 𝑀)
2916, 15zsubcld 12084 . . . . . . . . . . . 12 (𝜑 → (𝑀 − 1) ∈ ℤ)
30 fzn 12922 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
3116, 29, 30syl2anc 587 . . . . . . . . . . 11 (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
3228, 31mpbid 235 . . . . . . . . . 10 (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
3332adantr 484 . . . . . . . . 9 ((𝜑𝐼 = 𝑀) → (𝑀...(𝑀 − 1)) = ∅)
3433uneq2d 4093 . . . . . . . 8 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ ∅))
35 un0 4301 . . . . . . . . 9 ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1))
3635a1i 11 . . . . . . . 8 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)))
3734, 36eqtrd 2836 . . . . . . 7 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = (1...(𝑀 − 1)))
3837uneq1d 4092 . . . . . 6 ((𝜑𝐼 = 𝑀) → (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
3927, 38eqtrd 2836 . . . . 5 ((𝜑𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
4039eqcomd 2807 . . . 4 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
4115adantr 484 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℤ)
4216adantr 484 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℤ)
4342, 41zsubcld 12084 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 1) ∈ ℤ)
444nnzd 12078 . . . . . . 7 (𝜑𝐼 ∈ ℤ)
4544adantr 484 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℤ)
464nnge1d 11677 . . . . . . 7 (𝜑 → 1 ≤ 𝐼)
4746adantr 484 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ 𝐼)
48 eqid 2801 . . . . . . . . . . 11 𝑀 = 𝑀
49 eqeq1 2805 . . . . . . . . . . 11 (𝑀 = 𝐼 → (𝑀 = 𝑀𝐼 = 𝑀))
5048, 49mpbii 236 . . . . . . . . . 10 (𝑀 = 𝐼𝐼 = 𝑀)
5150necon3bi 3016 . . . . . . . . 9 𝐼 = 𝑀𝑀𝐼)
5251adantl 485 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀𝐼)
534nnred 11644 . . . . . . . . . 10 (𝜑𝐼 ∈ ℝ)
5453, 18, 5leltned 10786 . . . . . . . . 9 (𝜑 → (𝐼 < 𝑀𝑀𝐼))
5554adantr 484 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀𝑀𝐼))
5652, 55mpbird 260 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 < 𝑀)
57 zltlem1 12027 . . . . . . . . 9 ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
5844, 16, 57syl2anc 587 . . . . . . . 8 (𝜑 → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
5958adantr 484 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
6056, 59mpbid 235 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ≤ (𝑀 − 1))
6141, 43, 45, 47, 60fzsplitnr 39264 . . . . 5 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
6261uneq1d 4092 . . . 4 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
6340, 62pm2.61dan 812 . . 3 (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
64 fzsn 12948 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
6516, 64syl 17 . . . 4 (𝜑 → (𝑀...𝑀) = {𝑀})
6665uneq2d 4093 . . 3 (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
6721, 63, 663eqtrd 2840 . 2 (𝜑 → (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
68 uncom 4083 . . . . . 6 ((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1)))
6968a1i 11 . . . . 5 (𝜑 → ((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
7069uneq1d 4092 . . . 4 (𝜑 → (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
7165uneq2d 4093 . . . . . 6 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
7271eqcomd 2807 . . . . 5 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
73 fz10 12927 . . . . . . . . . . . . . . 15 (1...0) = ∅
7473uneq1i 4089 . . . . . . . . . . . . . 14 ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1)))
7574a1i 11 . . . . . . . . . . . . 13 (𝜑 → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
7675adantr 484 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
77 uncom 4083 . . . . . . . . . . . . . . 15 ((1...(𝑀 − 1)) ∪ ∅) = (∅ ∪ (1...(𝑀 − 1)))
7877eqeq1i 2806 . . . . . . . . . . . . . 14 (((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)) ↔ (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
7978imbi2i 339 . . . . . . . . . . . . 13 (((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1))) ↔ ((𝜑𝐼 = 𝑀) → (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1))))
8036, 79mpbi 233 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
8176, 80eqtrd 2836 . . . . . . . . . . 11 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
8281eqcomd 2807 . . . . . . . . . 10 ((𝜑𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...0) ∪ (1...(𝑀 − 1))))
83 oveq2 7147 . . . . . . . . . . . . . . 15 (𝐼 = 𝑀 → (𝑀𝐼) = (𝑀𝑀))
8483adantl 485 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = (𝑀𝑀))
8518recnd 10662 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℂ)
8685subidd 10978 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀𝑀) = 0)
8786adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝑀) = 0)
8884, 87eqtrd 2836 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = 0)
8988oveq2d 7155 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (1...(𝑀𝐼)) = (1...0))
9083oveq1d 7154 . . . . . . . . . . . . . . . 16 (𝐼 = 𝑀 → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9190adantl 485 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9287oveq1d 7154 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝑀) + 1) = (0 + 1))
9391, 92eqtrd 2836 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = (0 + 1))
94 1e0p1 12132 . . . . . . . . . . . . . 14 1 = (0 + 1)
9593, 94eqtr4di 2854 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = 1)
9695oveq1d 7154 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (((𝑀𝐼) + 1)...(𝑀 − 1)) = (1...(𝑀 − 1)))
9789, 96uneq12d 4094 . . . . . . . . . . 11 ((𝜑𝐼 = 𝑀) → ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) = ((1...0) ∪ (1...(𝑀 − 1))))
9897eqcomd 2807 . . . . . . . . . 10 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
9982, 98eqtrd 2836 . . . . . . . . 9 ((𝜑𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
10042, 45zsubcld 12084 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ ℤ)
10153adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℝ)
10218adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℝ)
103 1red 10635 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℝ)
104101, 102, 103, 60lesubd 11237 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ (𝑀𝐼))
105103, 101, 102, 47lesub2dd 11250 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ≤ (𝑀 − 1))
10641, 43, 100, 104, 105elfzd 12897 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ (1...(𝑀 − 1)))
107 fzsplit 12932 . . . . . . . . . 10 ((𝑀𝐼) ∈ (1...(𝑀 − 1)) → (1...(𝑀 − 1)) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
108106, 107syl 17 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
10999, 108pm2.61dan 812 . . . . . . . 8 (𝜑 → (1...(𝑀 − 1)) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
110109uneq1d 4092 . . . . . . 7 (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
11121, 110eqtrd 2836 . . . . . 6 (𝜑 → (1...𝑀) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
112111eqcomd 2807 . . . . 5 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (1...𝑀))
11372, 112eqtrd 2836 . . . 4 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (1...𝑀))
11470, 113eqtrd 2836 . . 3 (𝜑 → (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}) = (1...𝑀))
115114eqcomd 2807 . 2 (𝜑 → (1...𝑀) = (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}))
11614, 67, 1153jca 1125 1 (𝜑 → ((((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅ ∧ (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}) ∧ (1...𝑀) = (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  cun 3882  cin 3883  c0 4246  {csn 4528   class class class wbr 5033  (class class class)co 7139  cr 10529  0cc0 10530  1c1 10531   + caddc 10533   < clt 10668  cle 10669  cmin 10863  cn 11629  cz 11973  ...cfz 12889
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12382  df-fz 12890
This theorem is referenced by:  metakunt25  39365
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