![]() |
Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
|
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 13561 | . . . . . . 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 12638 | . . . . . . . 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 12638 | . . . . . . . 8 ⊢ (𝜑 → 𝐼 ∈ ℤ) |
8 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝐼 − 1))) → 𝐼 ∈ ℤ) |
9 | 5, 8 | zsubcld 12725 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...(𝐼 − 1))) → (𝑀 − 𝐼) ∈ ℤ) |
10 | 2, 9 | zaddcld 12724 | . . . . 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 6738 | . . 3 ⊢ (𝜑 → 𝐶 Fn (1...(𝐼 − 1))) |
14 | elfzelz 13561 | . . . . . . 7 ⊢ (𝑥 ∈ (𝐼...(𝑀 − 1)) → 𝑥 ∈ ℤ) | |
15 | 14 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝑥 ∈ ℤ) |
16 | 1zzd 12646 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 1 ∈ ℤ) | |
17 | 7 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → 𝐼 ∈ ℤ) |
18 | 16, 17 | zsubcld 12725 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐼...(𝑀 − 1))) → (1 − 𝐼) ∈ ℤ) |
19 | 15, 18 | zaddcld 12724 | . . . . 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 6738 | . . 3 ⊢ (𝜑 → 𝐷 Fn (𝐼...(𝑀 − 1))) |
23 | metakunt19.3 | . . . . . . 7 ⊢ (𝜑 → 𝐼 ≤ 𝑀) | |
24 | 3, 6, 23 | metakunt18 42204 | . . . . . 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 1141 | . . . 4 ⊢ (𝜑 → ((1...(𝐼 − 1)) ∩ (𝐼...(𝑀 − 1))) = ∅) |
27 | 13, 22, 26 | fnund 6684 | . . 3 ⊢ (𝜑 → (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1)))) |
28 | 13, 22, 27 | 3jca 1127 | . 2 ⊢ (𝜑 → (𝐶 Fn (1...(𝐼 − 1)) ∧ 𝐷 Fn (𝐼...(𝑀 − 1)) ∧ (𝐶 ∪ 𝐷) Fn ((1...(𝐼 − 1)) ∪ (𝐼...(𝑀 − 1))))) |
29 | fnsng 6620 | . . 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 1086 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ∩ cin 3962 ∅c0 4339 ifcif 4531 {csn 4631 〈cop 4637 class class class wbr 5148 ↦ cmpt 5231 Fn wfn 6558 (class class class)co 7431 1c1 11154 + caddc 11156 < clt 11293 ≤ cle 11294 − cmin 11490 ℕcn 12264 ℤcz 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: metakunt20 42206 metakunt21 42207 metakunt22 42208 metakunt25 42211 |
Copyright terms: Public domain | W3C validator |