fseq1m1p1 - Metamath Proof Explorer

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Theorem fseq1m1p1 13614

Description: Add/remove an item to/from the end of a finite sequence. (Contributed by Paul Chapman, 17-Nov-2012.)

**Hypothesis**
Ref	Expression
fseq1m1p1.1	⊢ 𝐻 = {⟨𝑁, 𝐵⟩}

**Assertion**
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))))))

**Proof of Theorem fseq1m1p1**
Step	Hyp	Ref	Expression
1		nnm1nn0 12540	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
2		eqid 2735	. . . 4 ⊢ {⟨((𝑁 − 1) + 1), 𝐵⟩} = {⟨((𝑁 − 1) + 1), 𝐵⟩}
3	2	fseq1p1m1 13613	. . 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 12246	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
6		ax-1cn 11185	. . . . . . . . 9 ⊢ 1 ∈ ℂ
7		npcan 11489	. . . . . . . . 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 4855	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ⟨((𝑁 − 1) + 1), 𝐵⟩ = ⟨𝑁, 𝐵⟩)
10	9	sneqd 4613	. . . . . 6 ⊢ (𝑁 ∈ ℕ → {⟨((𝑁 − 1) + 1), 𝐵⟩} = {⟨𝑁, 𝐵⟩})
11		fseq1m1p1.1	. . . . . 6 ⊢ 𝐻 = {⟨𝑁, 𝐵⟩}
12	10, 11	eqtr4di 2788	. . . . 5 ⊢ (𝑁 ∈ ℕ → {⟨((𝑁 − 1) + 1), 𝐵⟩} = 𝐻)
13	12	uneq2d 4143	. . . 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 1444	. 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 7419	. . . 4 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 − 1) + 1)) = (1...𝑁))
17	16	feq2d 6691	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ↔ 𝐺:(1...𝑁)⟶𝐴))
18	8	fveqeq2d 6883	. . 3 ⊢ (𝑁 ∈ ℕ → ((𝐺‘((𝑁 − 1) + 1)) = 𝐵 ↔ (𝐺‘𝑁) = 𝐵))
19	17, 18	3anbi12d 1439	. 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 1540 ∈ wcel 2108 ∪ cun 3924 {csn 4601 ⟨cop 4607 ↾ cres 5656 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 ℂcc 11125 1c1 11128 + caddc 11130 − cmin 11464 ℕcn 12238 ℕ₀cn0 12499 ...cfz 13522

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8717 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-n0 12500 df-z 12587 df-uz 12851 df-fz 13523

This theorem is referenced by: (None)

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