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Mirrors > Home > MPE Home > Th. List > fvmptnn04ifc | Structured version Visualization version GIF version |
Description: The function value of a mapping from the nonnegative integers with four distinct cases for the third 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 |
---|---|
fvmptnn04ifc | ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐶) |
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 1133 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → 𝑆 ∈ ℕ) |
4 | fvmptnn04if.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
5 | 4 | 3ad2ant1 1133 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → 𝑁 ∈ ℕ0) |
6 | simp3 1138 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) | |
7 | nnne0 12145 | . . . . . . . . 9 ⊢ (𝑆 ∈ ℕ → 𝑆 ≠ 0) | |
8 | 7 | neneqd 2946 | . . . . . . . 8 ⊢ (𝑆 ∈ ℕ → ¬ 𝑆 = 0) |
9 | 2, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → ¬ 𝑆 = 0) |
10 | 9 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → ¬ 𝑆 = 0) |
11 | eqeq1 2740 | . . . . . . . 8 ⊢ (𝑁 = 𝑆 → (𝑁 = 0 ↔ 𝑆 = 0)) | |
12 | 11 | notbid 317 | . . . . . . 7 ⊢ (𝑁 = 𝑆 → (¬ 𝑁 = 0 ↔ ¬ 𝑆 = 0)) |
13 | 12 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → (¬ 𝑁 = 0 ↔ ¬ 𝑆 = 0)) |
14 | 10, 13 | mpbird 256 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → ¬ 𝑁 = 0) |
15 | 14 | 3adant3 1132 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → ¬ 𝑁 = 0) |
16 | 15 | pm2.21d 121 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → (𝑁 = 0 → ⦋𝑁 / 𝑛⦌𝐶 = ⦋𝑁 / 𝑛⦌𝐴)) |
17 | 16 | imp 407 | . 2 ⊢ (((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐶 = ⦋𝑁 / 𝑛⦌𝐴) |
18 | 4 | nn0red 12432 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
19 | 2 | nnred 12126 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
20 | 18, 19 | lttri3d 11253 | . . . . . . 7 ⊢ (𝜑 → (𝑁 = 𝑆 ↔ (¬ 𝑁 < 𝑆 ∧ ¬ 𝑆 < 𝑁))) |
21 | 20 | simprbda 499 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → ¬ 𝑁 < 𝑆) |
22 | 21 | pm2.21d 121 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → (𝑁 < 𝑆 → ⦋𝑁 / 𝑛⦌𝐶 = ⦋𝑁 / 𝑛⦌𝐵)) |
23 | 22 | 3adant3 1132 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → (𝑁 < 𝑆 → ⦋𝑁 / 𝑛⦌𝐶 = ⦋𝑁 / 𝑛⦌𝐵)) |
24 | 23 | a1d 25 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → (0 < 𝑁 → (𝑁 < 𝑆 → ⦋𝑁 / 𝑛⦌𝐶 = ⦋𝑁 / 𝑛⦌𝐵))) |
25 | 24 | 3imp 1111 | . 2 ⊢ (((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 = ⦋𝑁 / 𝑛⦌𝐵) |
26 | eqidd 2737 | . 2 ⊢ (((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐶 = ⦋𝑁 / 𝑛⦌𝐶) | |
27 | 20 | simplbda 500 | . . . . 5 ⊢ ((𝜑 ∧ 𝑁 = 𝑆) → ¬ 𝑆 < 𝑁) |
28 | 27 | 3adant3 1132 | . . . 4 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → ¬ 𝑆 < 𝑁) |
29 | 28 | pm2.21d 121 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → (𝑆 < 𝑁 → ⦋𝑁 / 𝑛⦌𝐶 = ⦋𝑁 / 𝑛⦌𝐷)) |
30 | 29 | imp 407 | . 2 ⊢ (((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐶 = ⦋𝑁 / 𝑛⦌𝐷) |
31 | 1, 3, 5, 6, 17, 25, 26, 30 | fvmptnn04if 22144 | 1 ⊢ ((𝜑 ∧ 𝑁 = 𝑆 ∧ ⦋𝑁 / 𝑛⦌𝐶 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⦋csb 3853 ifcif 4484 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6493 0cc0 11009 < clt 11147 ℕcn 12111 ℕ0cn0 12371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-n0 12372 |
This theorem is referenced by: (None) |
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