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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 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))
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
StepHypRef Expression
1 fargshiftlem 26162 . . . . . . 7 ((𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁)) → (𝑙 + 1) ∈ (1...𝑁))
2 simpl 472 . . . . . . . . . . 11 (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) → (𝑙 + 1) ∈ (1...𝑁))
32adantr 480 . . . . . . . . . 10 ((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑙 + 1) ∈ (1...𝑁))
4 fveq2 6103 . . . . . . . . . . . . . 14 (𝑘 = (𝑙 + 1) → (𝐹𝑘) = (𝐹‘(𝑙 + 1)))
54fveq2d 6107 . . . . . . . . . . . . 13 (𝑘 = (𝑙 + 1) → (𝐸‘(𝐹𝑘)) = (𝐸‘(𝐹‘(𝑙 + 1))))
6 csbeq1 3502 . . . . . . . . . . . . 13 (𝑘 = (𝑙 + 1) → 𝑘 / 𝑥𝑃 = (𝑙 + 1) / 𝑥𝑃)
75, 6eqeq12d 2625 . . . . . . . . . . . 12 (𝑘 = (𝑙 + 1) → ((𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = (𝑙 + 1) / 𝑥𝑃))
87adantl 481 . . . . . . . . . . 11 (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 ↔ (𝐸‘(𝐹‘(𝑙 + 1))) = (𝑙 + 1) / 𝑥𝑃))
9 simpl 472 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁)) → 𝑁 ∈ ℕ0)
109adantl 481 . . . . . . . . . . . . . . . 16 (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) → 𝑁 ∈ ℕ0)
1110anim1i 590 . . . . . . . . . . . . . . 15 ((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸))
1211adantr 480 . . . . . . . . . . . . . 14 (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸))
13 simpr 476 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁)) → 𝑙 ∈ (0..^𝑁))
1413ad3antlr 763 . . . . . . . . . . . . . 14 (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → 𝑙 ∈ (0..^𝑁))
15 fargshift.g . . . . . . . . . . . . . . . . 17 𝐺 = (𝑥 ∈ (0..^(#‘𝐹)) ↦ (𝐹‘(𝑥 + 1)))
1615fargshiftfv 26163 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (𝑙 ∈ (0..^𝑁) → (𝐺𝑙) = (𝐹‘(𝑙 + 1))))
1716imp 444 . . . . . . . . . . . . . . 15 (((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑙 ∈ (0..^𝑁)) → (𝐺𝑙) = (𝐹‘(𝑙 + 1)))
1817eqcomd 2616 . . . . . . . . . . . . . 14 (((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑙 ∈ (0..^𝑁)) → (𝐹‘(𝑙 + 1)) = (𝐺𝑙))
1912, 14, 18syl2anc 691 . . . . . . . . . . . . 13 (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝐹‘(𝑙 + 1)) = (𝐺𝑙))
2019fveq2d 6107 . . . . . . . . . . . 12 (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → (𝐸‘(𝐹‘(𝑙 + 1))) = (𝐸‘(𝐺𝑙)))
2120eqeq1d 2612 . . . . . . . . . . 11 (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹‘(𝑙 + 1))) = (𝑙 + 1) / 𝑥𝑃 ↔ (𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))
228, 21bitrd 267 . . . . . . . . . 10 (((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) ∧ 𝑘 = (𝑙 + 1)) → ((𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 ↔ (𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))
233, 22rspcdv 3285 . . . . . . . . 9 ((((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) ∧ 𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → (𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))
2423ex 449 . . . . . . . 8 (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) → (𝐹:(1...𝑁)⟶dom 𝐸 → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → (𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃)))
2524com23 84 . . . . . . 7 (((𝑙 + 1) ∈ (1...𝑁) ∧ (𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁))) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → (𝐹:(1...𝑁)⟶dom 𝐸 → (𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃)))
261, 25mpancom 700 . . . . . 6 ((𝑁 ∈ ℕ0𝑙 ∈ (0..^𝑁)) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → (𝐹:(1...𝑁)⟶dom 𝐸 → (𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃)))
2726ex 449 . . . . 5 (𝑁 ∈ ℕ0 → (𝑙 ∈ (0..^𝑁) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → (𝐹:(1...𝑁)⟶dom 𝐸 → (𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))))
2827com24 93 . . . 4 (𝑁 ∈ ℕ0 → (𝐹:(1...𝑁)⟶dom 𝐸 → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → (𝑙 ∈ (0..^𝑁) → (𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))))
2928imp31 447 . . 3 (((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) ∧ ∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃) → (𝑙 ∈ (0..^𝑁) → (𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))
3029ralrimiv 2948 . 2 (((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) ∧ ∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃) → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃)
3130ex 449 1 ((𝑁 ∈ ℕ0𝐹:(1...𝑁)⟶dom 𝐸) → (∀𝑘 ∈ (1...𝑁)(𝐸‘(𝐹𝑘)) = 𝑘 / 𝑥𝑃 → ∀𝑙 ∈ (0..^𝑁)(𝐸‘(𝐺𝑙)) = (𝑙 + 1) / 𝑥𝑃))
 Colors of variables: wff setvar class 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)
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