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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 11211 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | eqid 2610 | . . . 4 ⊢ {〈((𝑁 − 1) + 1), 𝐵〉} = {〈((𝑁 − 1) + 1), 𝐵〉} | |
3 | 2 | fseq1p1m1 12283 | . . 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 10905 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
6 | ax-1cn 9873 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
7 | npcan 10169 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
8 | 5, 6, 7 | sylancl 693 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | opeq1d 4346 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 〈((𝑁 − 1) + 1), 𝐵〉 = 〈𝑁, 𝐵〉) |
10 | 9 | sneqd 4137 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = {〈𝑁, 𝐵〉}) |
11 | fseq1m1p1.1 | . . . . . 6 ⊢ 𝐻 = {〈𝑁, 𝐵〉} | |
12 | 10, 11 | syl6eqr 2662 | . . . . 5 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = 𝐻) |
13 | 12 | uneq2d 3729 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) = (𝐹 ∪ 𝐻)) |
14 | 13 | eqeq2d 2620 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺 = (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) ↔ 𝐺 = (𝐹 ∪ 𝐻))) |
15 | 14 | 3anbi3d 1397 | . 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 6565 | . . . 4 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
17 | 16 | feq2d 5944 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ↔ 𝐺:(1...𝑁)⟶𝐴)) |
18 | 8 | fveq2d 6107 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐺‘((𝑁 − 1) + 1)) = (𝐺‘𝑁)) |
19 | 18 | eqeq1d 2612 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝐺‘((𝑁 − 1) + 1)) = 𝐵 ↔ (𝐺‘𝑁) = 𝐵)) |
20 | 17, 19 | 3anbi12d 1392 | . 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 297 | 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 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∪ cun 3538 {csn 4125 〈cop 4131 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 1c1 9816 + caddc 9818 − cmin 10145 ℕcn 10897 ℕ0cn0 11169 ...cfz 12197 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 |
This theorem is referenced by: (None) |
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