Theorem fargshiftfva 41379

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,𝑘,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		fz0add1fz1 12537	. . . . . . 7 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁)) → (𝑙 + 1) ∈ (1...𝑁))
2		simpl 473	. . . . . . . . . . 11 ⊢ (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) → (𝑙 + 1) ∈ (1...𝑁))
3	2	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑙 + 1) ∈ (1...𝑁))
4		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑙 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑙 + 1)))
5	4	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑙 + 1) → (𝐸‘(𝐹‘𝑘)) = (𝐸‘(𝐹‘(𝑙 + 1))))
6		csbeq1 3536	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑙 + 1) → ⦋𝑘 / 𝑥⦌𝑃 = ⦋(𝑙 + 1) / 𝑥⦌𝑃)
7	5, 6	eqeq12d 2637	. . . . . . . . . . . 12 ⊢ (𝑘 = (𝑙 + 1) → ((𝐸‘(𝐹‘𝑘)) = ⦋𝑘 / 𝑥⦌𝑃 ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = ⦋(𝑙 + 1) / 𝑥⦌𝑃))
8	7	adantl 482	. . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
10	9	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁))) → 𝑁 ∈ ℕ0)
11	10	anim1i 592	. . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑙 ∈ (0..^𝑁)) → 𝑙 ∈ (0..^𝑁))
14	13	ad3antlr 767	. . . . . . . . . . . . . 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 41375	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑙 ∈ (0..^𝑁) → (𝐺‘𝑙) = (𝐹‘(𝑙 + 1))))
17	16	imp 445	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑙 ∈ (0..^𝑁)) → (𝐺‘𝑙) = (𝐹‘(𝑙 + 1)))
18	17	eqcomd 2628	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑙 ∈ (0..^𝑁)) → (𝐹‘(𝑙 + 1)) = (𝐺‘𝑙))
19	12, 14, 18	syl2anc 693	. . . . . . . . . . . . 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 6195	. . . . . . . . . . . 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 2624	. . . . . . . . . . 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 268	. . . . . . . . . 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 3312	. . . . . . . . 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 450	. . . . . . . 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 86	. . . . . . 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 703	. . . . . 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 450	. . . . 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 95	. . . 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 448	. . 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 2965	. 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 450	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ⦋csb 3533   ↦ cmpt 4729  dom cdm 5114  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939  ℕ₀cn0 11292  ...cfz 12326  ..^cfzo 12465  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118

This theorem is referenced by: (None)