itcovalt2lem2 - Mathbox for Alexander van der Vekens

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Theorem itcovalt2lem2 46915

Description: Lemma 2 for itcovalt2 46916: induction step. (Contributed by AV, 7-May-2024.)

**Hypothesis**
Ref	Expression
itcovalt2.f	⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 𝐶))

**Assertion**
Ref	Expression
itcovalt2lem2	⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))

Distinct variable groups: 𝐶,𝑛 𝑦,𝑛 Allowed substitution hints: 𝐶(𝑦) 𝐹(𝑦,𝑛)

Proof of Theorem itcovalt2lem2 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		itcovalt2.f	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 𝐶))
2		nn0ex 12443	. . . . . 6 ⊢ ℕ₀ ∈ V
3	2	mptex 7193	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 𝐶)) ∈ V
4	1, 3	eqeltri 2828	. . . 4 ⊢ 𝐹 ∈ V
5		simpl 483	. . . 4 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝑦 ∈ ℕ₀)
6		simpr 485	. . . 4 ⊢ (((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))
7		itcovalsucov 46907	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑦 ∈ ℕ₀ ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))))
8	4, 5, 6, 7	mp3an2ani 1468	. . 3 ⊢ (((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))))
9		2nn 12250	. . . . . . . . 9 ⊢ 2 ∈ ℕ
10	9	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℕ)
11		id 22	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℕ₀)
12	10, 11	nnexpcld 14173	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑𝑦) ∈ ℕ)
13		itcovalt2lem2lem1 46912	. . . . . . 7 ⊢ ((((2↑𝑦) ∈ ℕ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑛 ∈ ℕ₀) → (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) ∈ ℕ₀)
14	12, 13	sylanl1 678	. . . . . 6 ⊢ (((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑛 ∈ ℕ₀) → (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) ∈ ℕ₀)
15		eqidd 2732	. . . . . 6 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))
16		oveq2 7385	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (2 · 𝑛) = (2 · 𝑚))
17	16	oveq1d 7392	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ((2 · 𝑛) + 𝐶) = ((2 · 𝑚) + 𝐶))
18	17	cbvmptv 5238	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 𝐶)) = (𝑚 ∈ ℕ₀ ↦ ((2 · 𝑚) + 𝐶))
19	1, 18	eqtri 2759	. . . . . . 7 ⊢ 𝐹 = (𝑚 ∈ ℕ₀ ↦ ((2 · 𝑚) + 𝐶))
20	19	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝐹 = (𝑚 ∈ ℕ₀ ↦ ((2 · 𝑚) + 𝐶)))
21		oveq2 7385	. . . . . . 7 ⊢ (𝑚 = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) → (2 · 𝑚) = (2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))
22	21	oveq1d 7392	. . . . . 6 ⊢ (𝑚 = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) → ((2 · 𝑚) + 𝐶) = ((2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶))
23	14, 15, 20, 22	fmptco 7095	. . . . 5 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝐹 ∘ (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) = (𝑛 ∈ ℕ₀ ↦ ((2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶)))
24		itcovalt2lem2lem2 46913	. . . . . 6 ⊢ (((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑛 ∈ ℕ₀) → ((2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶) = (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))
25	24	mpteq2dva 5225	. . . . 5 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝑛 ∈ ℕ₀ ↦ ((2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶)) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))
26	23, 25	eqtrd 2771	. . . 4 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝐹 ∘ (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))
27	26	adantr 481	. . 3 ⊢ (((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → (𝐹 ∘ (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))
28	8, 27	eqtrd 2771	. 2 ⊢ (((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))
29	28	ex 413	1 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3459 ↦ cmpt 5208 ∘ ccom 5657 ‘cfv 6516 (class class class)co 7377 1c1 11076 + caddc 11078 · cmul 11080 − cmin 11409 ℕcn 12177 2c2 12232 ℕ₀cn0 12437 ↑cexp 13992 IterCompcitco 46896

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-2 12240 df-n0 12438 df-z 12524 df-uz 12788 df-seq 13932 df-exp 13993 df-itco 46898

This theorem is referenced by: itcovalt2 46916

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