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Mirrors > Home > MPE Home > Th. List > fvmptnn04ifa | Structured version Visualization version GIF version |
Description: The function value of a mapping from the nonnegative integers with four distinct cases for the first case. (Contributed by AV, 10-Nov-2019.) |
Ref | Expression |
---|---|
fvmptnn04if.g | ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵)))) |
fvmptnn04if.s | ⊢ (𝜑 → 𝑆 ∈ ℕ) |
fvmptnn04if.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
fvmptnn04ifa | ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptnn04if.g | . 2 ⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ if(𝑛 = 0, 𝐴, if(𝑛 = 𝑆, 𝐶, if(𝑆 < 𝑛, 𝐷, 𝐵)))) | |
2 | fvmptnn04if.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ ℕ) | |
3 | 2 | 3ad2ant1 1127 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → 𝑆 ∈ ℕ) |
4 | fvmptnn04if.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
5 | 4 | 3ad2ant1 1127 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → 𝑁 ∈ ℕ0) |
6 | simp3 1132 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) | |
7 | eqidd 2772 | . 2 ⊢ (((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐴 = ⦋𝑁 / 𝑛⦌𝐴) | |
8 | simpr 471 | . . . . . . . 8 ⊢ ((𝜑 ∧ 0 < 𝑁) → 0 < 𝑁) | |
9 | 8 | gt0ne0d 10794 | . . . . . . 7 ⊢ ((𝜑 ∧ 0 < 𝑁) → 𝑁 ≠ 0) |
10 | 9 | neneqd 2948 | . . . . . 6 ⊢ ((𝜑 ∧ 0 < 𝑁) → ¬ 𝑁 = 0) |
11 | 10 | pm2.21d 119 | . . . . 5 ⊢ ((𝜑 ∧ 0 < 𝑁) → (𝑁 = 0 → (𝑁 < 𝑆 → ⦋𝑁 / 𝑛⦌𝐴 = ⦋𝑁 / 𝑛⦌𝐵))) |
12 | 11 | impancom 439 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0) → (0 < 𝑁 → (𝑁 < 𝑆 → ⦋𝑁 / 𝑛⦌𝐴 = ⦋𝑁 / 𝑛⦌𝐵))) |
13 | 12 | 3adant3 1126 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → (0 < 𝑁 → (𝑁 < 𝑆 → ⦋𝑁 / 𝑛⦌𝐴 = ⦋𝑁 / 𝑛⦌𝐵))) |
14 | 13 | 3imp 1101 | . 2 ⊢ (((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → ⦋𝑁 / 𝑛⦌𝐴 = ⦋𝑁 / 𝑛⦌𝐵) |
15 | 2 | nnne0d 11267 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ≠ 0) |
16 | 15 | necomd 2998 | . . . . . . . 8 ⊢ (𝜑 → 0 ≠ 𝑆) |
17 | 16 | adantr 466 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 = 0) → 0 ≠ 𝑆) |
18 | neeq1 3005 | . . . . . . . 8 ⊢ (𝑁 = 0 → (𝑁 ≠ 𝑆 ↔ 0 ≠ 𝑆)) | |
19 | 18 | adantl 467 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 = 0) → (𝑁 ≠ 𝑆 ↔ 0 ≠ 𝑆)) |
20 | 17, 19 | mpbird 247 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 ≠ 𝑆) |
21 | 20 | 3adant3 1126 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → 𝑁 ≠ 𝑆) |
22 | 21 | neneqd 2948 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → ¬ 𝑁 = 𝑆) |
23 | 22 | pm2.21d 119 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → (𝑁 = 𝑆 → ⦋𝑁 / 𝑛⦌𝐴 = ⦋𝑁 / 𝑛⦌𝐶)) |
24 | 23 | imp 393 | . 2 ⊢ (((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐴 = ⦋𝑁 / 𝑛⦌𝐶) |
25 | nnnn0 11501 | . . . . . . . 8 ⊢ (𝑆 ∈ ℕ → 𝑆 ∈ ℕ0) | |
26 | nn0nlt0 11521 | . . . . . . . 8 ⊢ (𝑆 ∈ ℕ0 → ¬ 𝑆 < 0) | |
27 | 2, 25, 26 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑆 < 0) |
28 | 27 | adantr 466 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → ¬ 𝑆 < 0) |
29 | breq2 4790 | . . . . . . . 8 ⊢ (𝑁 = 0 → (𝑆 < 𝑁 ↔ 𝑆 < 0)) | |
30 | 29 | notbid 307 | . . . . . . 7 ⊢ (𝑁 = 0 → (¬ 𝑆 < 𝑁 ↔ ¬ 𝑆 < 0)) |
31 | 30 | adantl 467 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 0) → (¬ 𝑆 < 𝑁 ↔ ¬ 𝑆 < 0)) |
32 | 28, 31 | mpbird 247 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 0) → ¬ 𝑆 < 𝑁) |
33 | 32 | 3adant3 1126 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → ¬ 𝑆 < 𝑁) |
34 | 33 | pm2.21d 119 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → (𝑆 < 𝑁 → ⦋𝑁 / 𝑛⦌𝐴 = ⦋𝑁 / 𝑛⦌𝐷)) |
35 | 34 | imp 393 | . 2 ⊢ (((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐴 = ⦋𝑁 / 𝑛⦌𝐷) |
36 | 1, 3, 5, 6, 7, 14, 24, 35 | fvmptnn04if 20874 | 1 ⊢ ((𝜑 ∧ 𝑁 = 0 ∧ ⦋𝑁 / 𝑛⦌𝐴 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ⦋csb 3682 ifcif 4225 class class class wbr 4786 ↦ cmpt 4863 ‘cfv 6031 0cc0 10138 < clt 10276 ℕcn 11222 ℕ0cn0 11494 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-n0 11495 |
This theorem is referenced by: (None) |
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