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Theorem metakunt24 41008
Description: Technical condition such that metakunt17 41001 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 4276 . . . 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 41002 . . . . . . 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 496 . . . . . 6 (𝜑 → (((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅))
87simp2d 1144 . . . . 5 (𝜑 → ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅)
97simp3d 1145 . . . . 5 (𝜑 → ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅)
108, 9uneq12d 4165 . . . 4 (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = (∅ ∪ ∅))
11 unidm 4153 . . . . 5 (∅ ∪ ∅) = ∅
1211a1i 11 . . . 4 (𝜑 → (∅ ∪ ∅) = ∅)
1310, 12eqtrd 2773 . . 3 (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = ∅)
142, 13eqtrd 2773 . 2 (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
15 1zzd 12593 . . . . 5 (𝜑 → 1 ∈ ℤ)
163nnzd 12585 . . . . 5 (𝜑𝑀 ∈ ℤ)
173nnge1d 12260 . . . . 5 (𝜑 → 1 ≤ 𝑀)
183nnred 12227 . . . . . 6 (𝜑𝑀 ∈ ℝ)
1918leidd 11780 . . . . 5 (𝜑𝑀𝑀)
2015, 16, 16, 17, 19elfzd 13492 . . . 4 (𝜑𝑀 ∈ (1...𝑀))
2120fzsplitnd 40848 . . 3 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
22 oveq1 7416 . . . . . . . . . 10 (𝐼 = 𝑀 → (𝐼 − 1) = (𝑀 − 1))
2322oveq2d 7425 . . . . . . . . 9 (𝐼 = 𝑀 → (1...(𝐼 − 1)) = (1...(𝑀 − 1)))
24 oveq1 7416 . . . . . . . . 9 (𝐼 = 𝑀 → (𝐼...(𝑀 − 1)) = (𝑀...(𝑀 − 1)))
2523, 24uneq12d 4165 . . . . . . . 8 (𝐼 = 𝑀 → ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))))
2625uneq1d 4163 . . . . . . 7 (𝐼 = 𝑀 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
2726adantl 483 . . . . . 6 ((𝜑𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
2818ltm1d 12146 . . . . . . . . . . 11 (𝜑 → (𝑀 − 1) < 𝑀)
2916, 15zsubcld 12671 . . . . . . . . . . . 12 (𝜑 → (𝑀 − 1) ∈ ℤ)
30 fzn 13517 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
3116, 29, 30syl2anc 585 . . . . . . . . . . 11 (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
3228, 31mpbid 231 . . . . . . . . . 10 (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
3332adantr 482 . . . . . . . . 9 ((𝜑𝐼 = 𝑀) → (𝑀...(𝑀 − 1)) = ∅)
3433uneq2d 4164 . . . . . . . 8 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ ∅))
35 un0 4391 . . . . . . . . 9 ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1))
3635a1i 11 . . . . . . . 8 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)))
3734, 36eqtrd 2773 . . . . . . 7 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = (1...(𝑀 − 1)))
3837uneq1d 4163 . . . . . 6 ((𝜑𝐼 = 𝑀) → (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
3927, 38eqtrd 2773 . . . . 5 ((𝜑𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
4039eqcomd 2739 . . . 4 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
4115adantr 482 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℤ)
4216adantr 482 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℤ)
4342, 41zsubcld 12671 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 1) ∈ ℤ)
444nnzd 12585 . . . . . . 7 (𝜑𝐼 ∈ ℤ)
4544adantr 482 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℤ)
464nnge1d 12260 . . . . . . 7 (𝜑 → 1 ≤ 𝐼)
4746adantr 482 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ 𝐼)
48 eqid 2733 . . . . . . . . . . 11 𝑀 = 𝑀
49 eqeq1 2737 . . . . . . . . . . 11 (𝑀 = 𝐼 → (𝑀 = 𝑀𝐼 = 𝑀))
5048, 49mpbii 232 . . . . . . . . . 10 (𝑀 = 𝐼𝐼 = 𝑀)
5150necon3bi 2968 . . . . . . . . 9 𝐼 = 𝑀𝑀𝐼)
5251adantl 483 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀𝐼)
534nnred 12227 . . . . . . . . . 10 (𝜑𝐼 ∈ ℝ)
5453, 18, 5leltned 11367 . . . . . . . . 9 (𝜑 → (𝐼 < 𝑀𝑀𝐼))
5554adantr 482 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀𝑀𝐼))
5652, 55mpbird 257 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 < 𝑀)
57 zltlem1 12615 . . . . . . . . 9 ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
5844, 16, 57syl2anc 585 . . . . . . . 8 (𝜑 → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
5958adantr 482 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
6056, 59mpbid 231 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ≤ (𝑀 − 1))
6141, 43, 45, 47, 60fzsplitnr 40849 . . . . 5 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
6261uneq1d 4163 . . . 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 13543 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
6516, 64syl 17 . . . 4 (𝜑 → (𝑀...𝑀) = {𝑀})
6665uneq2d 4164 . . 3 (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
6721, 63, 663eqtrd 2777 . 2 (𝜑 → (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
68 uncom 4154 . . . . . 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 4163 . . . 4 (𝜑 → (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
7165uneq2d 4164 . . . . . 6 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
7271eqcomd 2739 . . . . 5 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
73 fz10 13522 . . . . . . . . . . . . . . 15 (1...0) = ∅
7473uneq1i 4160 . . . . . . . . . . . . . 14 ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1)))
7574a1i 11 . . . . . . . . . . . . 13 (𝜑 → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
7675adantr 482 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
77 uncom 4154 . . . . . . . . . . . . . . 15 ((1...(𝑀 − 1)) ∪ ∅) = (∅ ∪ (1...(𝑀 − 1)))
7877eqeq1i 2738 . . . . . . . . . . . . . 14 (((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)) ↔ (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
7978imbi2i 336 . . . . . . . . . . . . 13 (((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1))) ↔ ((𝜑𝐼 = 𝑀) → (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1))))
8036, 79mpbi 229 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
8176, 80eqtrd 2773 . . . . . . . . . . 11 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
8281eqcomd 2739 . . . . . . . . . 10 ((𝜑𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...0) ∪ (1...(𝑀 − 1))))
83 oveq2 7417 . . . . . . . . . . . . . . 15 (𝐼 = 𝑀 → (𝑀𝐼) = (𝑀𝑀))
8483adantl 483 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = (𝑀𝑀))
8518recnd 11242 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℂ)
8685subidd 11559 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀𝑀) = 0)
8786adantr 482 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝑀) = 0)
8884, 87eqtrd 2773 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = 0)
8988oveq2d 7425 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (1...(𝑀𝐼)) = (1...0))
9083oveq1d 7424 . . . . . . . . . . . . . . . 16 (𝐼 = 𝑀 → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9190adantl 483 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9287oveq1d 7424 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝑀) + 1) = (0 + 1))
9391, 92eqtrd 2773 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = (0 + 1))
94 1e0p1 12719 . . . . . . . . . . . . . 14 1 = (0 + 1)
9593, 94eqtr4di 2791 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = 1)
9695oveq1d 7424 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (((𝑀𝐼) + 1)...(𝑀 − 1)) = (1...(𝑀 − 1)))
9789, 96uneq12d 4165 . . . . . . . . . . 11 ((𝜑𝐼 = 𝑀) → ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) = ((1...0) ∪ (1...(𝑀 − 1))))
9897eqcomd 2739 . . . . . . . . . 10 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
9982, 98eqtrd 2773 . . . . . . . . 9 ((𝜑𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
10042, 45zsubcld 12671 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ ℤ)
10153adantr 482 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℝ)
10218adantr 482 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℝ)
103 1red 11215 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℝ)
104101, 102, 103, 60lesubd 11818 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ (𝑀𝐼))
105103, 101, 102, 47lesub2dd 11831 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ≤ (𝑀 − 1))
10641, 43, 100, 104, 105elfzd 13492 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ (1...(𝑀 − 1)))
107 fzsplit 13527 . . . . . . . . . 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 4163 . . . . . . 7 (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
11121, 110eqtrd 2773 . . . . . 6 (𝜑 → (1...𝑀) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
112111eqcomd 2739 . . . . 5 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (1...𝑀))
11372, 112eqtrd 2773 . . . 4 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (1...𝑀))
11470, 113eqtrd 2773 . . 3 (𝜑 → (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}) = (1...𝑀))
115114eqcomd 2739 . 2 (𝜑 → (1...𝑀) = (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}))
11614, 67, 1153jca 1129 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  cun 3947  cin 3948  c0 4323  {csn 4629   class class class wbr 5149  (class class class)co 7409  cr 11109  0cc0 11110  1c1 11111   + caddc 11113   < clt 11248  cle 11249  cmin 11444  cn 12212  cz 12558  ...cfz 13484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-rp 12975  df-fz 13485
This theorem is referenced by:  metakunt25  41009
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