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Mirrors > Home > MPE Home > Th. List > fseq1m1p1 | Structured version Visualization version 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 12017 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | eqid 2738 | . . . 4 ⊢ {〈((𝑁 − 1) + 1), 𝐵〉} = {〈((𝑁 − 1) + 1), 𝐵〉} | |
3 | 2 | fseq1p1m1 13072 | . . 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 17 | . 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 11724 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
6 | ax-1cn 10673 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
7 | npcan 10973 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
8 | 5, 6, 7 | sylancl 589 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | opeq1d 4767 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 〈((𝑁 − 1) + 1), 𝐵〉 = 〈𝑁, 𝐵〉) |
10 | 9 | sneqd 4528 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = {〈𝑁, 𝐵〉}) |
11 | fseq1m1p1.1 | . . . . . 6 ⊢ 𝐻 = {〈𝑁, 𝐵〉} | |
12 | 10, 11 | eqtr4di 2791 | . . . . 5 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = 𝐻) |
13 | 12 | uneq2d 4053 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) = (𝐹 ∪ 𝐻)) |
14 | 13 | eqeq2d 2749 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺 = (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) ↔ 𝐺 = (𝐹 ∪ 𝐻))) |
15 | 14 | 3anbi3d 1443 | . 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 7186 | . . . 4 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
17 | 16 | feq2d 6490 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ↔ 𝐺:(1...𝑁)⟶𝐴)) |
18 | 8 | fveqeq2d 6682 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝐺‘((𝑁 − 1) + 1)) = 𝐵 ↔ (𝐺‘𝑁) = 𝐵)) |
19 | 17, 18 | 3anbi12d 1438 | . 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)))))) |
20 | 4, 15, 19 | 3bitr3d 312 | 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 setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∪ cun 3841 {csn 4516 〈cop 4522 ↾ cres 5527 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ℂcc 10613 1c1 10616 + caddc 10618 − cmin 10948 ℕcn 11716 ℕ0cn0 11976 ...cfz 12981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 |
This theorem is referenced by: (None) |
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