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Theorem metakunt24 40646
Description: Technical condition such that metakunt17 40639 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 4236 . . . 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 40640 . . . . . . 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 4125 . . . 4 (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = (∅ ∪ ∅))
11 unidm 4113 . . . . 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 12539 . . . . 5 (𝜑 → 1 ∈ ℤ)
163nnzd 12531 . . . . 5 (𝜑𝑀 ∈ ℤ)
173nnge1d 12206 . . . . 5 (𝜑 → 1 ≤ 𝑀)
183nnred 12173 . . . . . 6 (𝜑𝑀 ∈ ℝ)
1918leidd 11726 . . . . 5 (𝜑𝑀𝑀)
2015, 16, 16, 17, 19elfzd 13438 . . . 4 (𝜑𝑀 ∈ (1...𝑀))
2120fzsplitnd 40486 . . 3 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
22 oveq1 7365 . . . . . . . . . 10 (𝐼 = 𝑀 → (𝐼 − 1) = (𝑀 − 1))
2322oveq2d 7374 . . . . . . . . 9 (𝐼 = 𝑀 → (1...(𝐼 − 1)) = (1...(𝑀 − 1)))
24 oveq1 7365 . . . . . . . . 9 (𝐼 = 𝑀 → (𝐼...(𝑀 − 1)) = (𝑀...(𝑀 − 1)))
2523, 24uneq12d 4125 . . . . . . . 8 (𝐼 = 𝑀 → ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))))
2625uneq1d 4123 . . . . . . 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 12092 . . . . . . . . . . 11 (𝜑 → (𝑀 − 1) < 𝑀)
2916, 15zsubcld 12617 . . . . . . . . . . . 12 (𝜑 → (𝑀 − 1) ∈ ℤ)
30 fzn 13463 . . . . . . . . . . . 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 4124 . . . . . . . 8 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ ∅))
35 un0 4351 . . . . . . . . 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 4123 . . . . . 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 12617 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 1) ∈ ℤ)
444nnzd 12531 . . . . . . 7 (𝜑𝐼 ∈ ℤ)
4544adantr 482 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℤ)
464nnge1d 12206 . . . . . . 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 2967 . . . . . . . . 9 𝐼 = 𝑀𝑀𝐼)
5251adantl 483 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀𝐼)
534nnred 12173 . . . . . . . . . 10 (𝜑𝐼 ∈ ℝ)
5453, 18, 5leltned 11313 . . . . . . . . 9 (𝜑 → (𝐼 < 𝑀𝑀𝐼))
5554adantr 482 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀𝑀𝐼))
5652, 55mpbird 257 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 < 𝑀)
57 zltlem1 12561 . . . . . . . . 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 40487 . . . . 5 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
6261uneq1d 4123 . . . 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 13489 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
6516, 64syl 17 . . . 4 (𝜑 → (𝑀...𝑀) = {𝑀})
6665uneq2d 4124 . . 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 4114 . . . . . 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 4123 . . . 4 (𝜑 → (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
7165uneq2d 4124 . . . . . 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 13468 . . . . . . . . . . . . . . 15 (1...0) = ∅
7473uneq1i 4120 . . . . . . . . . . . . . 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 4114 . . . . . . . . . . . . . . 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 7366 . . . . . . . . . . . . . . 15 (𝐼 = 𝑀 → (𝑀𝐼) = (𝑀𝑀))
8483adantl 483 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = (𝑀𝑀))
8518recnd 11188 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℂ)
8685subidd 11505 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀𝑀) = 0)
8786adantr 482 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝑀) = 0)
8884, 87eqtrd 2773 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = 0)
8988oveq2d 7374 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (1...(𝑀𝐼)) = (1...0))
9083oveq1d 7373 . . . . . . . . . . . . . . . 16 (𝐼 = 𝑀 → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9190adantl 483 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9287oveq1d 7373 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝑀) + 1) = (0 + 1))
9391, 92eqtrd 2773 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = (0 + 1))
94 1e0p1 12665 . . . . . . . . . . . . . 14 1 = (0 + 1)
9593, 94eqtr4di 2791 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = 1)
9695oveq1d 7373 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (((𝑀𝐼) + 1)...(𝑀 − 1)) = (1...(𝑀 − 1)))
9789, 96uneq12d 4125 . . . . . . . . . . 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 12617 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ ℤ)
10153adantr 482 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℝ)
10218adantr 482 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℝ)
103 1red 11161 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℝ)
104101, 102, 103, 60lesubd 11764 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ (𝑀𝐼))
105103, 101, 102, 47lesub2dd 11777 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ≤ (𝑀 − 1))
10641, 43, 100, 104, 105elfzd 13438 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ (1...(𝑀 − 1)))
107 fzsplit 13473 . . . . . . . . . 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 4123 . . . . . . 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 2940  cun 3909  cin 3910  c0 4283  {csn 4587   class class class wbr 5106  (class class class)co 7358  cr 11055  0cc0 11056  1c1 11057   + caddc 11059   < clt 11194  cle 11195  cmin 11390  cn 12158  cz 12504  ...cfz 13430
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-fz 13431
This theorem is referenced by:  metakunt25  40647
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