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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 11941 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0) | |
2 | eqid 2823 | . . . 4 ⊢ {〈((𝑁 − 1) + 1), 𝐵〉} = {〈((𝑁 − 1) + 1), 𝐵〉} | |
3 | 2 | fseq1p1m1 12984 | . . 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 11648 | . . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
6 | ax-1cn 10597 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
7 | npcan 10897 | . . . . . . . . 9 ⊢ ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 − 1) + 1) = 𝑁) | |
8 | 5, 6, 7 | sylancl 588 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝑁 − 1) + 1) = 𝑁) |
9 | 8 | opeq1d 4811 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 〈((𝑁 − 1) + 1), 𝐵〉 = 〈𝑁, 𝐵〉) |
10 | 9 | sneqd 4581 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = {〈𝑁, 𝐵〉}) |
11 | fseq1m1p1.1 | . . . . . 6 ⊢ 𝐻 = {〈𝑁, 𝐵〉} | |
12 | 10, 11 | syl6eqr 2876 | . . . . 5 ⊢ (𝑁 ∈ ℕ → {〈((𝑁 − 1) + 1), 𝐵〉} = 𝐻) |
13 | 12 | uneq2d 4141 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) = (𝐹 ∪ 𝐻)) |
14 | 13 | eqeq2d 2834 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺 = (𝐹 ∪ {〈((𝑁 − 1) + 1), 𝐵〉}) ↔ 𝐺 = (𝐹 ∪ 𝐻))) |
15 | 14 | 3anbi3d 1438 | . 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 7174 | . . . 4 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
17 | 16 | feq2d 6502 | . . 3 ⊢ (𝑁 ∈ ℕ → (𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ↔ 𝐺:(1...𝑁)⟶𝐴)) |
18 | 8 | fveqeq2d 6680 | . . 3 ⊢ (𝑁 ∈ ℕ → ((𝐺‘((𝑁 − 1) + 1)) = 𝐵 ↔ (𝐺‘𝑁) = 𝐵)) |
19 | 17, 18 | 3anbi12d 1433 | . 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 311 | 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 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 {csn 4569 〈cop 4575 ↾ cres 5559 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 1c1 10540 + caddc 10542 − cmin 10872 ℕcn 11640 ℕ0cn0 11900 ...cfz 12895 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 |
This theorem is referenced by: (None) |
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