Theorem fargshiftfva 26167

Description: The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)

Hypothesis

Ref	Expression
fargshift.g	⊢ 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))

Assertion

Ref	Expression
fargshiftfva	⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐸   𝑘,𝐹,𝑙,𝑥   𝑥,𝑁   𝑘,𝐸   𝑘,𝐺   𝑘,𝑁   𝑃,𝑘   𝐸,𝑙   𝑁,𝑙   𝑃,𝑙

Allowed substitution hints:   𝑃(𝑥)   𝐺(𝑥,𝑙)

Proof of Theorem fargshiftfva

Step	Hyp	Ref	Expression
1		fargshiftlem 26162	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁)) → (𝑙 + 1) ∈ (1...𝑁))
2		simpl 472	. . . . . . . . . . 11 ⊢ (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) → (𝑙 + 1) ∈ (1...𝑁))
3	2	adantr 480	. . . . . . . . . 10 ⊢ ((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑙 + 1) ∈ (1...𝑁))
4		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑙 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑙 + 1)))
5	4	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑙 + 1) → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘(𝑙 + 1))))
6		csbeq1 3502	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑙 + 1) → ⦋𝑘 / 𝑥⦌𝑃 = ⦋(𝑙 + 1) / 𝑥⦌𝑃)
7	5, 6	eqeq12d 2625	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑙 + 1) → ((𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))
8	7	adantl 481	. . . . . . . . . . 11 ⊢ (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))
9		simpl 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
10	9	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) → 𝑁 ∈ ℕ0)
11	10	anim1i 590	. . . . . . . . . . . . . . 15 ⊢ ((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸))
12	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸))
13		simpr 476	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁)) → 𝑙 ∈ (0..^𝑁))
14	13	ad3antlr 763	. . . . . . . . . . . . . 14 ⊢ (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → 𝑙 ∈ (0..^𝑁))
15		fargshift.g	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
16	15	fargshiftfv 26163	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑙 ∈ (0..^𝑁) → (𝐺‘𝑙) = (𝐹‘(𝑙 + 1))))
17	16	imp 444	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑙 ∈ (0..^𝑁)) → (𝐺‘𝑙) = (𝐹‘(𝑙 + 1)))
18	17	eqcomd 2616	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑙 ∈ (0..^𝑁)) → (𝐹‘(𝑙 + 1)) = (𝐺‘𝑙))
19	12, 14, 18	syl2anc 691	. . . . . . . . . . . . 13 ⊢ (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝐹‘(𝑙 + 1)) = (𝐺‘𝑙))
20	19	fveq2d 6107	. . . . . . . . . . . 12 ⊢ (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝐸‘(𝐹‘(𝑙 + 1))) = (𝐸‘(𝐺‘𝑙)))
21	20	eqeq1d 2612	. . . . . . . . . . 11 ⊢ (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹‘(𝑙 + 1))) = ⦋(𝑙 + 1) / 𝑥⦌𝑃 ↔ (𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))
22	8, 21	bitrd 267	. . . . . . . . . 10 ⊢ (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 ↔ (𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))
23	3, 22	rspcdv 3285	. . . . . . . . 9 ⊢ ((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → (𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))
24	23	ex 449	. . . . . . . 8 ⊢ (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) → (𝐹:(1...𝑁)⟶dom 𝐸 → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → (𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃)))
25	24	com23 84	. . . . . . 7 ⊢ (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → (𝐹:(1...𝑁)⟶dom 𝐸 → (𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃)))
26	1, 25	mpancom 700	. . . . . 6 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁)) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → (𝐹:(1...𝑁)⟶dom 𝐸 → (𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃)))
27	26	ex 449	. . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑙 ∈ (0..^𝑁) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → (𝐹:(1...𝑁)⟶dom 𝐸 → (𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))))
28	27	com24 93	. . . 4 ⊢ (𝑁 ∈ ℕ0 → (𝐹:(1...𝑁)⟶dom 𝐸 → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → (𝑙 ∈ (0..^𝑁) → (𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))))
29	28	imp31 447	. . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ ∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃) → (𝑙 ∈ (0..^𝑁) → (𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))
30	29	ralrimiv 2948	. 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ ∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃) → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃)
31	30	ex 449	1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺‘𝑙)) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ⦋csb 3499   ↦ cmpt 4643  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕ₀cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979

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-rep 4699  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-int 4411  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-1o 7447  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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  df-fzo 12335  df-hash 12980

This theorem is referenced by: (None)