![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fvmptnn04ifb | Structured version Visualization version GIF version |
Description: The function value of a mapping from the nonnegative integers with four distinct cases for the second 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 |
---|---|
fvmptnn04ifb | ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐵) |
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 1131 | . 2 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → 𝑆 ∈ ℕ) |
4 | fvmptnn04if.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
5 | 4 | 3ad2ant1 1131 | . 2 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → 𝑁 ∈ ℕ0) |
6 | simp3 1136 | . 2 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) | |
7 | nn0re 12497 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℝ) | |
8 | nn0ge0 12513 | . . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ0 → 0 ≤ 𝑁) | |
9 | 7, 8 | jca 511 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℝ ∧ 0 ≤ 𝑁)) |
10 | ne0gt0 11335 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 0 ≤ 𝑁) → (𝑁 ≠ 0 ↔ 0 < 𝑁)) | |
11 | 4, 9, 10 | 3syl 18 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 ≠ 0 ↔ 0 < 𝑁)) |
12 | 11 | biimprcd 249 | . . . . . . 7 ⊢ (0 < 𝑁 → (𝜑 → 𝑁 ≠ 0)) |
13 | 12 | adantr 480 | . . . . . 6 ⊢ ((0 < 𝑁 ∧ 𝑁 < 𝑆) → (𝜑 → 𝑁 ≠ 0)) |
14 | 13 | impcom 407 | . . . . 5 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆)) → 𝑁 ≠ 0) |
15 | 14 | 3adant3 1130 | . . . 4 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → 𝑁 ≠ 0) |
16 | neneq 2941 | . . . . 5 ⊢ (𝑁 ≠ 0 → ¬ 𝑁 = 0) | |
17 | 16 | pm2.21d 121 | . . . 4 ⊢ (𝑁 ≠ 0 → (𝑁 = 0 → ⦋𝑁 / 𝑛⦌𝐵 = ⦋𝑁 / 𝑛⦌𝐴)) |
18 | 15, 17 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → (𝑁 = 0 → ⦋𝑁 / 𝑛⦌𝐵 = ⦋𝑁 / 𝑛⦌𝐴)) |
19 | 18 | imp 406 | . 2 ⊢ (((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) ∧ 𝑁 = 0) → ⦋𝑁 / 𝑛⦌𝐵 = ⦋𝑁 / 𝑛⦌𝐴) |
20 | eqidd 2728 | . 2 ⊢ (((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) ∧ 0 < 𝑁 ∧ 𝑁 < 𝑆) → ⦋𝑁 / 𝑛⦌𝐵 = ⦋𝑁 / 𝑛⦌𝐵) | |
21 | 4, 7 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
22 | 21 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 < 𝑆) → 𝑁 ∈ ℝ) |
23 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 < 𝑆) → 𝑁 < 𝑆) | |
24 | 22, 23 | ltned 11366 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 < 𝑆) → 𝑁 ≠ 𝑆) |
25 | 24 | neneqd 2940 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑁 < 𝑆) → ¬ 𝑁 = 𝑆) |
26 | 25 | adantrl 715 | . . . . 5 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆)) → ¬ 𝑁 = 𝑆) |
27 | 26 | 3adant3 1130 | . . . 4 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → ¬ 𝑁 = 𝑆) |
28 | 27 | pm2.21d 121 | . . 3 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → (𝑁 = 𝑆 → ⦋𝑁 / 𝑛⦌𝐵 = ⦋𝑁 / 𝑛⦌𝐶)) |
29 | 28 | imp 406 | . 2 ⊢ (((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) ∧ 𝑁 = 𝑆) → ⦋𝑁 / 𝑛⦌𝐵 = ⦋𝑁 / 𝑛⦌𝐶) |
30 | 2 | nnred 12243 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℝ) |
31 | ltnsym 11328 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑁 < 𝑆 → ¬ 𝑆 < 𝑁)) | |
32 | 21, 30, 31 | syl2anc 583 | . . . . . . . 8 ⊢ (𝜑 → (𝑁 < 𝑆 → ¬ 𝑆 < 𝑁)) |
33 | 32 | com12 32 | . . . . . . 7 ⊢ (𝑁 < 𝑆 → (𝜑 → ¬ 𝑆 < 𝑁)) |
34 | 33 | adantl 481 | . . . . . 6 ⊢ ((0 < 𝑁 ∧ 𝑁 < 𝑆) → (𝜑 → ¬ 𝑆 < 𝑁)) |
35 | 34 | impcom 407 | . . . . 5 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆)) → ¬ 𝑆 < 𝑁) |
36 | 35 | 3adant3 1130 | . . . 4 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → ¬ 𝑆 < 𝑁) |
37 | 36 | pm2.21d 121 | . . 3 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → (𝑆 < 𝑁 → ⦋𝑁 / 𝑛⦌𝐵 = ⦋𝑁 / 𝑛⦌𝐷)) |
38 | 37 | imp 406 | . 2 ⊢ (((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) ∧ 𝑆 < 𝑁) → ⦋𝑁 / 𝑛⦌𝐵 = ⦋𝑁 / 𝑛⦌𝐷) |
39 | 1, 3, 5, 6, 19, 20, 29, 38 | fvmptnn04if 22725 | 1 ⊢ ((𝜑 ∧ (0 < 𝑁 ∧ 𝑁 < 𝑆) ∧ ⦋𝑁 / 𝑛⦌𝐵 ∈ 𝑉) → (𝐺‘𝑁) = ⦋𝑁 / 𝑛⦌𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ⦋csb 3889 ifcif 4524 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 ℝcr 11123 0cc0 11124 < clt 11264 ≤ cle 11265 ℕcn 12228 ℕ0cn0 12488 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-n0 12489 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |