![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 12565 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | eqid 2735 | . . . 4 ⊢ {〈((𝑁 − 1) + 1), 𝐵〉} = {〈((𝑁 − 1) + 1), 𝐵〉} | |
3 | 2 | fseq1p1m1 13635 | . . 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 12272 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
6 | ax-1cn 11211 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
7 | npcan 11515 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
8 | 5, 6, 7 | sylancl 586 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | opeq1d 4884 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 〈((𝑁 − 1) + 1), 𝐵〉 = 〈𝑁, 𝐵〉) |
10 | 9 | sneqd 4643 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = {〈𝑁, 𝐵〉}) |
11 | fseq1m1p1.1 | . . . . . 6 ⊢ 𝐻 = {〈𝑁, 𝐵〉} | |
12 | 10, 11 | eqtr4di 2793 | . . . . 5 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = 𝐻) |
13 | 12 | uneq2d 4178 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) = (𝐹 ∪ 𝐻)) |
14 | 13 | eqeq2d 2746 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺 = (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) ↔ 𝐺 = (𝐹 ∪ 𝐻))) |
15 | 14 | 3anbi3d 1441 | . 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 7447 | . . . 4 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
17 | 16 | feq2d 6723 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ↔ 𝐺:(1...𝑁)⟶𝐴)) |
18 | 8 | fveqeq2d 6915 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝐺‘((𝑁 − 1) + 1)) = 𝐵 ↔ (𝐺‘𝑁) = 𝐵)) |
19 | 17, 18 | 3anbi12d 1436 | . 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 309 | 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 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 {csn 4631 〈cop 4637 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 + caddc 11156 − cmin 11490 ℕcn 12264 ℕ0cn0 12524 ...cfz 13544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |