fseq1m1p1 - Metamath Proof Explorer

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Theorem fseq1m1p1 13517

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 12444	. . 3 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
2		eqid 2736	. . . 4 ⊢ {⟨((𝑁 − 1) + 1), 𝐵⟩} = {⟨((𝑁 − 1) + 1), 𝐵⟩}
3	2	fseq1p1m1 13516	. . 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 12155	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
6		ax-1cn 11086	. . . . . . . . 9 ⊢ 1 ∈ ℂ
7		npcan 11391	. . . . . . . . 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 4835	. . . . . . 7 ⊢ (𝑁 ∈ ℕ → ⟨((𝑁 − 1) + 1), 𝐵⟩ = ⟨𝑁, 𝐵⟩)
10	9	sneqd 4592	. . . . . 6 ⊢ (𝑁 ∈ ℕ → {⟨((𝑁 − 1) + 1), 𝐵⟩} = {⟨𝑁, 𝐵⟩})
11		fseq1m1p1.1	. . . . . 6 ⊢ 𝐻 = {⟨𝑁, 𝐵⟩}
12	10, 11	eqtr4di 2789	. . . . 5 ⊢ (𝑁 ∈ ℕ → {⟨((𝑁 − 1) + 1), 𝐵⟩} = 𝐻)
13	12	uneq2d 4120	. . . 4 ⊢ (𝑁 ∈ ℕ → (𝐹 ∪ {⟨((𝑁 − 1) + 1), 𝐵⟩}) = (𝐹 ∪ 𝐻))
14	13	eqeq2d 2747	. . 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 7374	. . . 4 ⊢ (𝑁 ∈ ℕ → (1...((𝑁 − 1) + 1)) = (1...𝑁))
17	16	feq2d 6646	. . 3 ⊢ (𝑁 ∈ ℕ → (𝐺:(1...((𝑁 − 1) + 1))⟶𝐴 ↔ 𝐺:(1...𝑁)⟶𝐴))
18	8	fveqeq2d 6842	. . 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 1541 ∈ wcel 2113 ∪ cun 3899 {csn 4580 ⟨cop 4586 ↾ cres 5626 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 1c1 11029 + caddc 11031 − cmin 11366 ℕcn 12147 ℕ₀cn0 12403 ...cfz 13425

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-er 8635 df-en 8886 df-dom 8887 df-sdom 8888 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-nn 12148 df-n0 12404 df-z 12491 df-uz 12754 df-fz 13426

This theorem is referenced by: (None)

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