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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 13498 | . . . . . . 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 12582 | . . . . . . . 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 12582 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝐼 − 1))) → 𝐼 ∈ ℤ) |
9 | 5, 8 | zsubcld 12668 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝐼 − 1))) → (𝑀 − 𝐼) ∈ ℤ) |
10 | 2, 9 | zaddcld 12667 | . . . . 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 7105 | . . . 4 ⊢ (𝜑 → 𝐶:(1...(𝐼 − 1))⟶ℤ) |
13 | 12 | ffnd 6708 | . . 3 ⊢ (𝜑 → 𝐶 Fn (1...(𝐼 − 1))) |
14 | elfzelz 13498 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐼...(𝑀 − 1)) → 𝑥 ∈ ℤ) | |
15 | 14 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝑥 ∈ ℤ) |
16 | 1zzd 12590 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 1 ∈ ℤ) | |
17 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝐼 ∈ ℤ) |
18 | 16, 17 | zsubcld 12668 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (1 − 𝐼) ∈ ℤ) |
19 | 15, 18 | zaddcld 12667 | . . . . 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 7105 | . . . 4 ⊢ (𝜑 → 𝐷:(𝐼...(𝑀 − 1))⟶ℤ) |
22 | 21 | ffnd 6708 | . . 3 ⊢ (𝜑 → 𝐷 Fn (𝐼...(𝑀 − 1))) |
23 | metakunt19.3 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
24 | 3, 6, 23 | metakunt18 41495 | . . . . . 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 1139 | . . . 4 ⊢ (𝜑 → ((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅) |
27 | 13, 22, 26 | fnund 6654 | . . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) |
28 | 13, 22, 27 | 3jca 1125 | . 2 ⊢ (𝜑 → (𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))) |
29 | fnsng 6590 | . . 3 ⊢ ((𝑀 ∈ ℕ ∧ 𝑀 ∈ ℕ) → {〈𝑀, 𝑀〉} Fn {𝑀}) | |
30 | 3, 3, 29 | syl2anc 583 | . 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 1084 = wceq 1533 ∈ wcel 2098 ∪ cun 3938 ∩ cin 3939 ∅c0 4314 ifcif 4520 {csn 4620 〈cop 4626 class class class wbr 5138 ↦ cmpt 5221 Fn wfn 6528 (class class class)co 7401 1c1 11107 + caddc 11109 < clt 11245 ≤ cle 11246 − cmin 11441 ℕcn 12209 ℤcz 12555 ...cfz 13481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fz 13482 |
This theorem is referenced by: metakunt20 41497 metakunt21 41498 metakunt22 41499 metakunt25 41502 |
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