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Theorem metakunt24 42210
Description: Technical condition such that metakunt17 42203 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 4292 . . . 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 42204 . . . . . . 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 494 . . . . . 6 (𝜑 → (((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅))
87simp2d 1142 . . . . 5 (𝜑 → ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅)
97simp3d 1143 . . . . 5 (𝜑 → ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅)
108, 9uneq12d 4179 . . . 4 (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = (∅ ∪ ∅))
11 unidm 4167 . . . . 5 (∅ ∪ ∅) = ∅
1211a1i 11 . . . 4 (𝜑 → (∅ ∪ ∅) = ∅)
1310, 12eqtrd 2775 . . 3 (𝜑 → (((1...(𝐼 − 1)) ∩ {𝑀}) ∪ ((𝐼...(𝑀 − 1)) ∩ {𝑀})) = ∅)
142, 13eqtrd 2775 . 2 (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∩ {𝑀}) = ∅)
15 1zzd 12646 . . . . 5 (𝜑 → 1 ∈ ℤ)
163nnzd 12638 . . . . 5 (𝜑𝑀 ∈ ℤ)
173nnge1d 12312 . . . . 5 (𝜑 → 1 ≤ 𝑀)
183nnred 12279 . . . . . 6 (𝜑𝑀 ∈ ℝ)
1918leidd 11827 . . . . 5 (𝜑𝑀𝑀)
2015, 16, 16, 17, 19elfzd 13552 . . . 4 (𝜑𝑀 ∈ (1...𝑀))
2120fzsplitnd 41964 . . 3 (𝜑 → (1...𝑀) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
22 oveq1 7438 . . . . . . . . . 10 (𝐼 = 𝑀 → (𝐼 − 1) = (𝑀 − 1))
2322oveq2d 7447 . . . . . . . . 9 (𝐼 = 𝑀 → (1...(𝐼 − 1)) = (1...(𝑀 − 1)))
24 oveq1 7438 . . . . . . . . 9 (𝐼 = 𝑀 → (𝐼...(𝑀 − 1)) = (𝑀...(𝑀 − 1)))
2523, 24uneq12d 4179 . . . . . . . 8 (𝐼 = 𝑀 → ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))))
2625uneq1d 4177 . . . . . . 7 (𝐼 = 𝑀 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
2726adantl 481 . . . . . 6 ((𝜑𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)))
2818ltm1d 12198 . . . . . . . . . . 11 (𝜑 → (𝑀 − 1) < 𝑀)
2916, 15zsubcld 12725 . . . . . . . . . . . 12 (𝜑 → (𝑀 − 1) ∈ ℤ)
30 fzn 13577 . . . . . . . . . . . 12 ((𝑀 ∈ ℤ ∧ (𝑀 − 1) ∈ ℤ) → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
3116, 29, 30syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((𝑀 − 1) < 𝑀 ↔ (𝑀...(𝑀 − 1)) = ∅))
3228, 31mpbid 232 . . . . . . . . . 10 (𝜑 → (𝑀...(𝑀 − 1)) = ∅)
3332adantr 480 . . . . . . . . 9 ((𝜑𝐼 = 𝑀) → (𝑀...(𝑀 − 1)) = ∅)
3433uneq2d 4178 . . . . . . . 8 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = ((1...(𝑀 − 1)) ∪ ∅))
35 un0 4400 . . . . . . . . 9 ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1))
3635a1i 11 . . . . . . . 8 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ ∅) = (1...(𝑀 − 1)))
3734, 36eqtrd 2775 . . . . . . 7 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) = (1...(𝑀 − 1)))
3837uneq1d 4177 . . . . . 6 ((𝜑𝐼 = 𝑀) → (((1...(𝑀 − 1)) ∪ (𝑀...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
3927, 38eqtrd 2775 . . . . 5 ((𝜑𝐼 = 𝑀) → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)))
4039eqcomd 2741 . . . 4 ((𝜑𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
4115adantr 480 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℤ)
4216adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℤ)
4342, 41zsubcld 12725 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀 − 1) ∈ ℤ)
444nnzd 12638 . . . . . . 7 (𝜑𝐼 ∈ ℤ)
4544adantr 480 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℤ)
464nnge1d 12312 . . . . . . 7 (𝜑 → 1 ≤ 𝐼)
4746adantr 480 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ 𝐼)
48 eqid 2735 . . . . . . . . . . 11 𝑀 = 𝑀
49 eqeq1 2739 . . . . . . . . . . 11 (𝑀 = 𝐼 → (𝑀 = 𝑀𝐼 = 𝑀))
5048, 49mpbii 233 . . . . . . . . . 10 (𝑀 = 𝐼𝐼 = 𝑀)
5150necon3bi 2965 . . . . . . . . 9 𝐼 = 𝑀𝑀𝐼)
5251adantl 481 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀𝐼)
534nnred 12279 . . . . . . . . . 10 (𝜑𝐼 ∈ ℝ)
5453, 18, 5leltned 11412 . . . . . . . . 9 (𝜑 → (𝐼 < 𝑀𝑀𝐼))
5554adantr 480 . . . . . . . 8 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀𝑀𝐼))
5652, 55mpbird 257 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 < 𝑀)
57 zltlem1 12668 . . . . . . . . 9 ((𝐼 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
5844, 16, 57syl2anc 584 . . . . . . . 8 (𝜑 → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
5958adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝐼 < 𝑀𝐼 ≤ (𝑀 − 1)))
6056, 59mpbid 232 . . . . . 6 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ≤ (𝑀 − 1))
6141, 43, 45, 47, 60fzsplitnr 41965 . . . . 5 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))
6261uneq1d 4177 . . . 4 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
6340, 62pm2.61dan 813 . . 3 (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)))
64 fzsn 13603 . . . . 5 (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
6516, 64syl 17 . . . 4 (𝜑 → (𝑀...𝑀) = {𝑀})
6665uneq2d 4178 . . 3 (𝜑 → (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
6721, 63, 663eqtrd 2779 . 2 (𝜑 → (1...𝑀) = (((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))) ∪ {𝑀}))
68 uncom 4168 . . . . . 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 4177 . . . 4 (𝜑 → (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
7165uneq2d 4178 . . . . . 6 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}))
7271eqcomd 2741 . . . . 5 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
73 fz10 13582 . . . . . . . . . . . . . . 15 (1...0) = ∅
7473uneq1i 4174 . . . . . . . . . . . . . 14 ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1)))
7574a1i 11 . . . . . . . . . . . . 13 (𝜑 → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
7675adantr 480 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (∅ ∪ (1...(𝑀 − 1))))
77 uncom 4168 . . . . . . . . . . . . . . 15 ((1...(𝑀 − 1)) ∪ ∅) = (∅ ∪ (1...(𝑀 − 1)))
7877eqeq1i 2740 . . . . . . . . . . . . . 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 230 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (∅ ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
8176, 80eqtrd 2775 . . . . . . . . . . 11 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = (1...(𝑀 − 1)))
8281eqcomd 2741 . . . . . . . . . 10 ((𝜑𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...0) ∪ (1...(𝑀 − 1))))
83 oveq2 7439 . . . . . . . . . . . . . . 15 (𝐼 = 𝑀 → (𝑀𝐼) = (𝑀𝑀))
8483adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = (𝑀𝑀))
8518recnd 11287 . . . . . . . . . . . . . . . 16 (𝜑𝑀 ∈ ℂ)
8685subidd 11606 . . . . . . . . . . . . . . 15 (𝜑 → (𝑀𝑀) = 0)
8786adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → (𝑀𝑀) = 0)
8884, 87eqtrd 2775 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → (𝑀𝐼) = 0)
8988oveq2d 7447 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (1...(𝑀𝐼)) = (1...0))
9083oveq1d 7446 . . . . . . . . . . . . . . . 16 (𝐼 = 𝑀 → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9190adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = ((𝑀𝑀) + 1))
9287oveq1d 7446 . . . . . . . . . . . . . . 15 ((𝜑𝐼 = 𝑀) → ((𝑀𝑀) + 1) = (0 + 1))
9391, 92eqtrd 2775 . . . . . . . . . . . . . 14 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = (0 + 1))
94 1e0p1 12773 . . . . . . . . . . . . . 14 1 = (0 + 1)
9593, 94eqtr4di 2793 . . . . . . . . . . . . 13 ((𝜑𝐼 = 𝑀) → ((𝑀𝐼) + 1) = 1)
9695oveq1d 7446 . . . . . . . . . . . 12 ((𝜑𝐼 = 𝑀) → (((𝑀𝐼) + 1)...(𝑀 − 1)) = (1...(𝑀 − 1)))
9789, 96uneq12d 4179 . . . . . . . . . . 11 ((𝜑𝐼 = 𝑀) → ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) = ((1...0) ∪ (1...(𝑀 − 1))))
9897eqcomd 2741 . . . . . . . . . 10 ((𝜑𝐼 = 𝑀) → ((1...0) ∪ (1...(𝑀 − 1))) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
9982, 98eqtrd 2775 . . . . . . . . 9 ((𝜑𝐼 = 𝑀) → (1...(𝑀 − 1)) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
10042, 45zsubcld 12725 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ ℤ)
10153adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝐼 ∈ ℝ)
10218adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 𝑀 ∈ ℝ)
103 1red 11260 . . . . . . . . . . . 12 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ∈ ℝ)
104101, 102, 103, 60lesubd 11865 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → 1 ≤ (𝑀𝐼))
105103, 101, 102, 47lesub2dd 11878 . . . . . . . . . . 11 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ≤ (𝑀 − 1))
10641, 43, 100, 104, 105elfzd 13552 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝐼 = 𝑀) → (𝑀𝐼) ∈ (1...(𝑀 − 1)))
107 fzsplit 13587 . . . . . . . . . 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 813 . . . . . . . 8 (𝜑 → (1...(𝑀 − 1)) = ((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))))
110109uneq1d 4177 . . . . . . 7 (𝜑 → ((1...(𝑀 − 1)) ∪ (𝑀...𝑀)) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
11121, 110eqtrd 2775 . . . . . 6 (𝜑 → (1...𝑀) = (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)))
112111eqcomd 2741 . . . . 5 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ (𝑀...𝑀)) = (1...𝑀))
11372, 112eqtrd 2775 . . . 4 (𝜑 → (((1...(𝑀𝐼)) ∪ (((𝑀𝐼) + 1)...(𝑀 − 1))) ∪ {𝑀}) = (1...𝑀))
11470, 113eqtrd 2775 . . 3 (𝜑 → (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}) = (1...𝑀))
115114eqcomd 2741 . 2 (𝜑 → (1...𝑀) = (((((𝑀𝐼) + 1)...(𝑀 − 1)) ∪ (1...(𝑀𝐼))) ∪ {𝑀}))
11614, 67, 1153jca 1127 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  cun 3961  cin 3962  c0 4339  {csn 4631   class class class wbr 5148  (class class class)co 7431  cr 11152  0cc0 11153  1c1 11154   + caddc 11156   < clt 11293  cle 11294  cmin 11490  cn 12264  cz 12611  ...cfz 13544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545
This theorem is referenced by:  metakunt25  42211
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