| Mathbox for metakunt |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > metakunt19 | Structured version Visualization version GIF version | ||
| Description: Domains on restrictions of functions. (Contributed by metakunt, 28-May-2024.) |
| Ref | Expression |
|---|---|
| metakunt19.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
| metakunt19.2 | ⊢ (𝜑 → 𝐼 ∈ ℕ) |
| metakunt19.3 | ⊢ (𝜑 → 𝐼 ≤ 𝑀) |
| metakunt19.4 | ⊢ 𝐵 = (𝑥 ∈ (1...𝑀) ↦ if(𝑥 = 𝑀, 𝑀, if(𝑥 < 𝐼, (𝑥 + (𝑀 − 𝐼)), (𝑥 + (1 − 𝐼))))) |
| metakunt19.5 | ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) |
| metakunt19.6 | ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) |
| Ref | Expression |
|---|---|
| metakunt19 | ⊢ (𝜑 → ((𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) ∧ {〈𝑀, 𝑀〉} Fn {𝑀})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elfzelz 13564 | . . . . . . 7 ⊢ (𝑥 ∈ (1...(𝐼 − 1)) → 𝑥 ∈ ℤ) | |
| 2 | 1 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝐼 − 1))) → 𝑥 ∈ ℤ) |
| 3 | metakunt19.1 | . . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
| 4 | 3 | nnzd 12640 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 5 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝐼 − 1))) → 𝑀 ∈ ℤ) |
| 6 | metakunt19.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ ℕ) | |
| 7 | 6 | nnzd 12640 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
| 8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝐼 − 1))) → 𝐼 ∈ ℤ) |
| 9 | 5, 8 | zsubcld 12727 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝐼 − 1))) → (𝑀 − 𝐼) ∈ ℤ) |
| 10 | 2, 9 | zaddcld 12726 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝐼 − 1))) → (𝑥 + (𝑀 − 𝐼)) ∈ ℤ) |
| 11 | metakunt19.5 | . . . . 5 ⊢ 𝐶 = (𝑥 ∈ (1...(𝐼 − 1)) ↦ (𝑥 + (𝑀 − 𝐼))) | |
| 12 | 10, 11 | fmptd 7134 | . . . 4 ⊢ (𝜑 → 𝐶:(1...(𝐼 − 1))⟶ℤ) |
| 13 | 12 | ffnd 6737 | . . 3 ⊢ (𝜑 → 𝐶 Fn (1...(𝐼 − 1))) |
| 14 | elfzelz 13564 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐼...(𝑀 − 1)) → 𝑥 ∈ ℤ) | |
| 15 | 14 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝑥 ∈ ℤ) |
| 16 | 1zzd 12648 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 1 ∈ ℤ) | |
| 17 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝐼 ∈ ℤ) |
| 18 | 16, 17 | zsubcld 12727 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (1 − 𝐼) ∈ ℤ) |
| 19 | 15, 18 | zaddcld 12726 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (𝑥 + (1 − 𝐼)) ∈ ℤ) |
| 20 | metakunt19.6 | . . . . 5 ⊢ 𝐷 = (𝑥 ∈ (𝐼...(𝑀 − 1)) ↦ (𝑥 + (1 − 𝐼))) | |
| 21 | 19, 20 | fmptd 7134 | . . . 4 ⊢ (𝜑 → 𝐷:(𝐼...(𝑀 − 1))⟶ℤ) |
| 22 | 21 | ffnd 6737 | . . 3 ⊢ (𝜑 → 𝐷 Fn (𝐼...(𝑀 − 1))) |
| 23 | metakunt19.3 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
| 24 | 3, 6, 23 | metakunt18 42223 | . . . . . 6 ⊢ (𝜑 → ((((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅) ∧ (((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∩ (1...(𝑀 − 𝐼))) = ∅ ∧ ((((𝑀 − 𝐼) + 1)...(𝑀 − 1)) ∩ {𝑀}) = ∅ ∧ ((1...(𝑀 − 𝐼)) ∩ {𝑀}) = ∅))) |
| 25 | 24 | simpld 494 | . . . . 5 ⊢ (𝜑 → (((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅ ∧ ((1...(𝐼 − 1)) ∩ {𝑀}) = ∅ ∧ ((𝐼...(𝑀 − 1)) ∩ {𝑀}) = ∅)) |
| 26 | 25 | simp1d 1143 | . . . 4 ⊢ (𝜑 → ((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅) |
| 27 | 13, 22, 26 | fnund 6683 | . . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) |
| 28 | 13, 22, 27 | 3jca 1129 | . 2 ⊢ (𝜑 → (𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))) |
| 29 | fnsng 6618 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑀 ∈ ℕ) → {〈𝑀, 𝑀〉} Fn {𝑀}) | |
| 30 | 3, 3, 29 | syl2anc 584 | . 2 ⊢ (𝜑 → {〈𝑀, 𝑀〉} Fn {𝑀}) |
| 31 | 28, 30 | jca 511 | 1 ⊢ (𝜑 → ((𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) ∧ {〈𝑀, 𝑀〉} Fn {𝑀})) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 ifcif 4525 {csn 4626 〈cop 4632 class class class wbr 5143 ↦ cmpt 5225 Fn wfn 6556 (class class class)co 7431 1c1 11156 + caddc 11158 < clt 11295 ≤ cle 11296 − cmin 11492 ℕcn 12266 ℤcz 12613 ...cfz 13547 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-fz 13548 |
| This theorem is referenced by: metakunt20 42225 metakunt21 42226 metakunt22 42227 metakunt25 42230 |
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