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Mirrors > Home > ILE Home > Th. List > fseq1m1p1 | GIF version |
Description: Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.) |
Ref | Expression |
---|---|
fseq1m1p1.1 | ⊢ 𝐻 = {〈𝑁, 𝐵〉} |
Ref | Expression |
---|---|
fseq1m1p1 | ⊢ (𝑁 ∈ ℕ → ((𝐹:(1...(𝑁 − 1))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...𝑁)⟶𝐴 ∧ (𝐺‘𝑁) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnm1nn0 9018 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | eqid 2139 | . . . 4 ⊢ {〈((𝑁 − 1) + 1), 𝐵〉} = {〈((𝑁 − 1) + 1), 𝐵〉} | |
3 | 2 | fseq1p1m1 9874 | . . 3 ⊢ ((𝑁 − 1) ∈ ℕ0 → ((𝐹:(1...(𝑁 − 1))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉})) ↔ (𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ∧ (𝐺‘((𝑁 − 1) + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1)))))) |
4 | 1, 3 | syl 14 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝐹:(1...(𝑁 − 1))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉})) ↔ (𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ∧ (𝐺‘((𝑁 − 1) + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1)))))) |
5 | nncn 8728 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
6 | ax-1cn 7713 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
7 | npcan 7971 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
8 | 5, 6, 7 | sylancl 409 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | opeq1d 3711 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 〈((𝑁 − 1) + 1), 𝐵〉 = 〈𝑁, 𝐵〉) |
10 | 9 | sneqd 3540 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = {〈𝑁, 𝐵〉}) |
11 | fseq1m1p1.1 | . . . . . 6 ⊢ 𝐻 = {〈𝑁, 𝐵〉} | |
12 | 10, 11 | syl6eqr 2190 | . . . . 5 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = 𝐻) |
13 | 12 | uneq2d 3230 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) = (𝐹 ∪ 𝐻)) |
14 | 13 | eqeq2d 2151 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺 = (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) ↔ 𝐺 = (𝐹 ∪ 𝐻))) |
15 | 14 | 3anbi3d 1296 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝐹:(1...(𝑁 − 1))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉})) ↔ (𝐹:(1...(𝑁 − 1))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)))) |
16 | 8 | oveq2d 5790 | . . . 4 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
17 | 16 | feq2d 5260 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ↔ 𝐺:(1...𝑁)⟶𝐴)) |
18 | 8 | fveq2d 5425 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐺‘((𝑁 − 1) + 1)) = (𝐺‘𝑁)) |
19 | 18 | eqeq1d 2148 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝐺‘((𝑁 − 1) + 1)) = 𝐵 ↔ (𝐺‘𝑁) = 𝐵)) |
20 | 17, 19 | 3anbi12d 1291 | . 2 ⊢ (𝑁 ∈ ℕ → ((𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ∧ (𝐺‘((𝑁 − 1) + 1)) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1)))) ↔ (𝐺:(1...𝑁)⟶𝐴 ∧ (𝐺‘𝑁) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1)))))) |
21 | 4, 15, 20 | 3bitr3d 217 | 1 ⊢ (𝑁 ∈ ℕ → ((𝐹:(1...(𝑁 − 1))⟶𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐺 = (𝐹 ∪ 𝐻)) ↔ (𝐺:(1...𝑁)⟶𝐴 ∧ (𝐺‘𝑁) = 𝐵 ∧ 𝐹 = (𝐺 ↾ (1...(𝑁 − 1)))))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 ∪ cun 3069 {csn 3527 〈cop 3530 ↾ cres 4541 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ℂcc 7618 1c1 7621 + caddc 7623 − cmin 7933 ℕcn 8720 ℕ0cn0 8977 ...cfz 9790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-fz 9791 |
This theorem is referenced by: (None) |
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