itcovalt2lem2 - Mathbox for Alexander van der Vekens

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Theorem itcovalt2lem2 47449

Description: Lemma 2 for itcovalt2 47450: 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 12482	. . . . . 6 ⊢ ℕ₀ ∈ V
3	2	mptex 7226	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 𝐶)) ∈ V
4	1, 3	eqeltri 2827	. . . 4 ⊢ 𝐹 ∈ V
5		simpl 481	. . . 4 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝑦 ∈ ℕ₀)
6		simpr 483	. . . 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 47441	. . . 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 1466	. . 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 12289	. . . . . . . . 9 ⊢ 2 ∈ ℕ
10	9	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → 2 ∈ ℕ)
11		id 22	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℕ₀)
12	10, 11	nnexpcld 14212	. . . . . . 7 ⊢ (𝑦 ∈ ℕ₀ → (2↑𝑦) ∈ ℕ)
13		itcovalt2lem2lem1 47446	. . . . . . 7 ⊢ ((((2↑𝑦) ∈ ℕ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑛 ∈ ℕ₀) → (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) ∈ ℕ₀)
14	12, 13	sylanl1 676	. . . . . 6 ⊢ (((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑛 ∈ ℕ₀) → (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) ∈ ℕ₀)
15		eqidd 2731	. . . . . 6 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))
16		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (2 · 𝑛) = (2 · 𝑚))
17	16	oveq1d 7426	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ((2 · 𝑛) + 𝐶) = ((2 · 𝑚) + 𝐶))
18	17	cbvmptv 5260	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ₀ ↦ ((2 · 𝑛) + 𝐶)) = (𝑚 ∈ ℕ₀ ↦ ((2 · 𝑚) + 𝐶))
19	1, 18	eqtri 2758	. . . . . . 7 ⊢ 𝐹 = (𝑚 ∈ ℕ₀ ↦ ((2 · 𝑚) + 𝐶))
20	19	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → 𝐹 = (𝑚 ∈ ℕ₀ ↦ ((2 · 𝑚) + 𝐶)))
21		oveq2 7419	. . . . . . 7 ⊢ (𝑚 = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) → (2 · 𝑚) = (2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)))
22	21	oveq1d 7426	. . . . . 6 ⊢ (𝑚 = (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶) → ((2 · 𝑚) + 𝐶) = ((2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶))
23	14, 15, 20, 22	fmptco 7128	. . . . 5 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝐹 ∘ (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) = (𝑛 ∈ ℕ₀ ↦ ((2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶)))
24		itcovalt2lem2lem2 47447	. . . . . 6 ⊢ (((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) ∧ 𝑛 ∈ ℕ₀) → ((2 · (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶)) + 𝐶) = (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶))
25	24	mpteq2dva 5247	. . . . 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 2770	. . . 4 ⊢ ((𝑦 ∈ ℕ₀ ∧ 𝐶 ∈ ℕ₀) → (𝐹 ∘ (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑𝑦)) − 𝐶))) = (𝑛 ∈ ℕ₀ ↦ (((𝑛 + 𝐶) · (2↑(𝑦 + 1))) − 𝐶)))
27	26	adantr 479	. . 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 2770	. 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 411	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 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ↦ cmpt 5230 ∘ ccom 5679 ‘cfv 6542 (class class class)co 7411 1c1 11113 + caddc 11115 · cmul 11117 − cmin 11448 ℕcn 12216 2c2 12271 ℕ₀cn0 12476 ↑cexp 14031 IterCompcitco 47430

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-seq 13971 df-exp 14032 df-itco 47432

This theorem is referenced by: itcovalt2 47450

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