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Theorem metakunt24 41314
Description: Technical condition such that metakunt17 41307 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 4274 . . . 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 41308 . . . . . . 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 493 . . . . . 6 (𝜑 → (((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅))
87simp2d 1141 . . . . 5 (𝜑 → ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅)
97simp3d 1142 . . . . 5 (𝜑 → ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅)
108, 9uneq12d 4163 . . . 4 (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = (∅ ∪ ∅))
11 unidm 4151 . . . . 5 (∅ ∪ ∅) = ∅
1211a1i 11 . . . 4 (𝜑 → (∅ ∪ ∅) = ∅)
1310, 12eqtrd 2770 . . 3 (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = ∅)
142, 13eqtrd 2770 . 2 (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
15 1zzd 12597 . . . . 5 (𝜑 → 1 ∈ ℤ)
163nnzd 12589 . . . . 5 (𝜑𝑀 ∈ ℤ)
173nnge1d 12264 . . . . 5 (𝜑 → 1 ≤ 𝑀)
183nnred 12231 . . . . . 6 (𝜑𝑀 ∈ ℝ)
1918leidd 11784 . . . . 5 (𝜑𝑀𝑀)
2015, 16, 16, 17, 19elfzd 13496 . . . 4 (𝜑𝑀 ∈ (1...𝑀))
2120fzsplitnd 41154 . . 3 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
22 oveq1 7418 . . . . . . . . . 10 (𝐼 = 𝑀 → (𝐼 − 1) = (𝑀 − 1))
2322oveq2d 7427 . . . . . . . . 9 (𝐼 = 𝑀 → (1...(𝐼 − 1)) = (1...(𝑀 − 1)))
24 oveq1 7418 . . . . . . . . 9 (𝐼 = 𝑀 → (𝐼...(𝑀 − 1)) = (𝑀...(𝑀 − 1)))
2523, 24uneq12d 4163 . . . . . . . 8 (𝐼 = 𝑀 → ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))))
2625uneq1d 4161 . . . . . . 7 (𝐼 = 𝑀 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
2726adantl 480 . . . . . 6 ((𝜑𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
2818ltm1d 12150 . . . . . . . . . . 11 (𝜑 → (𝑀 − 1) < 𝑀)
2916, 15zsubcld 12675 . . . . . . . . . . . 12 (𝜑 → (𝑀 − 1) ∈ ℤ)
30 fzn 13521 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
3116, 29, 30syl2anc 582 . . . . . . . . . . 11 (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
3228, 31mpbid 231 . . . . . . . . . 10 (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
3332adantr 479 . . . . . . . . 9 ((𝜑𝐼 = 𝑀) → (𝑀...(𝑀 − 1)) = ∅)
3433uneq2d 4162 . . . . . . . 8 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ ∅))
35 un0 4389 . . . . . . . . 9 ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1))
3635a1i 11 . . . . . . . 8 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)))
3734, 36eqtrd 2770 . . . . . . 7 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = (1...(𝑀 − 1)))
3837uneq1d 4161 . . . . . 6 ((𝜑𝐼 = 𝑀) → (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
3927, 38eqtrd 2770 . . . . 5 ((𝜑𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
4039eqcomd 2736 . . . 4 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
4115adantr 479 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℤ)
4216adantr 479 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℤ)
4342, 41zsubcld 12675 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 1) ∈ ℤ)
444nnzd 12589 . . . . . . 7 (𝜑𝐼 ∈ ℤ)
4544adantr 479 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℤ)
464nnge1d 12264 . . . . . . 7 (𝜑 → 1 ≤ 𝐼)
4746adantr 479 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ 𝐼)
48 eqid 2730 . . . . . . . . . . 11 𝑀 = 𝑀
49 eqeq1 2734 . . . . . . . . . . 11 (𝑀 = 𝐼 → (𝑀 = 𝑀𝐼 = 𝑀))
5048, 49mpbii 232 . . . . . . . . . 10 (𝑀 = 𝐼𝐼 = 𝑀)
5150necon3bi 2965 . . . . . . . . 9 𝐼 = 𝑀𝑀𝐼)
5251adantl 480 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀𝐼)
534nnred 12231 . . . . . . . . . 10 (𝜑𝐼 ∈ ℝ)
5453, 18, 5leltned 11371 . . . . . . . . 9 (𝜑 → (𝐼 < 𝑀𝑀𝐼))
5554adantr 479 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀𝑀𝐼))
5652, 55mpbird 256 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 < 𝑀)
57 zltlem1 12619 . . . . . . . . 9 ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
5844, 16, 57syl2anc 582 . . . . . . . 8 (𝜑 → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
5958adantr 479 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
6056, 59mpbid 231 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ≤ (𝑀 − 1))
6141, 43, 45, 47, 60fzsplitnr 41155 . . . . 5 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
6261uneq1d 4161 . . . 4 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
6340, 62pm2.61dan 809 . . 3 (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
64 fzsn 13547 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
6516, 64syl 17 . . . 4 (𝜑 → (𝑀...𝑀) = {𝑀})
6665uneq2d 4162 . . 3 (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
6721, 63, 663eqtrd 2774 . 2 (𝜑 → (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
68 uncom 4152 . . . . . 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 4161 . . . 4 (𝜑 → (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
7165uneq2d 4162 . . . . . 6 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
7271eqcomd 2736 . . . . 5 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
73 fz10 13526 . . . . . . . . . . . . . . 15 (1...0) = ∅
7473uneq1i 4158 . . . . . . . . . . . . . 14 ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1)))
7574a1i 11 . . . . . . . . . . . . 13 (𝜑 → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
7675adantr 479 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
77 uncom 4152 . . . . . . . . . . . . . . 15 ((1...(𝑀 − 1)) ∪ ∅) = (∅ ∪ (1...(𝑀 − 1)))
7877eqeq1i 2735 . . . . . . . . . . . . . 14 (((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)) ↔ (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
7978imbi2i 335 . . . . . . . . . . . . 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 2770 . . . . . . . . . . 11 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
8281eqcomd 2736 . . . . . . . . . 10 ((𝜑𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...0) ∪ (1...(𝑀 − 1))))
83 oveq2 7419 . . . . . . . . . . . . . . 15 (𝐼 = 𝑀 → (𝑀𝐼) = (𝑀𝑀))
8483adantl 480 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = (𝑀𝑀))
8518recnd 11246 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℂ)
8685subidd 11563 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀𝑀) = 0)
8786adantr 479 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝑀) = 0)
8884, 87eqtrd 2770 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = 0)
8988oveq2d 7427 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (1...(𝑀𝐼)) = (1...0))
9083oveq1d 7426 . . . . . . . . . . . . . . . 16 (𝐼 = 𝑀 → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9190adantl 480 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9287oveq1d 7426 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝑀) + 1) = (0 + 1))
9391, 92eqtrd 2770 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = (0 + 1))
94 1e0p1 12723 . . . . . . . . . . . . . 14 1 = (0 + 1)
9593, 94eqtr4di 2788 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = 1)
9695oveq1d 7426 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (((𝑀𝐼) + 1)...(𝑀 − 1)) = (1...(𝑀 − 1)))
9789, 96uneq12d 4163 . . . . . . . . . . 11 ((𝜑𝐼 = 𝑀) → ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) = ((1...0) ∪ (1...(𝑀 − 1))))
9897eqcomd 2736 . . . . . . . . . 10 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
9982, 98eqtrd 2770 . . . . . . . . 9 ((𝜑𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
10042, 45zsubcld 12675 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ ℤ)
10153adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℝ)
10218adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℝ)
103 1red 11219 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℝ)
104101, 102, 103, 60lesubd 11822 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ (𝑀𝐼))
105103, 101, 102, 47lesub2dd 11835 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ≤ (𝑀 − 1))
10641, 43, 100, 104, 105elfzd 13496 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ (1...(𝑀 − 1)))
107 fzsplit 13531 . . . . . . . . . 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 809 . . . . . . . 8 (𝜑 → (1...(𝑀 − 1)) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
110109uneq1d 4161 . . . . . . 7 (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
11121, 110eqtrd 2770 . . . . . 6 (𝜑 → (1...𝑀) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
112111eqcomd 2736 . . . . 5 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (1...𝑀))
11372, 112eqtrd 2770 . . . 4 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (1...𝑀))
11470, 113eqtrd 2770 . . 3 (𝜑 → (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}) = (1...𝑀))
115114eqcomd 2736 . 2 (𝜑 → (1...𝑀) = (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}))
11614, 67, 1153jca 1126 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 394  w3a 1085   = wceq 1539  wcel 2104  wne 2938  cun 3945  cin 3946  c0 4321  {csn 4627   class class class wbr 5147  (class class class)co 7411  cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252  cle 11253  cmin 11448  cn 12216  cz 12562  ...cfz 13488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-fz 13489
This theorem is referenced by:  metakunt25  41315
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