Theorem itcovalpclem2 49508

Description: Lemma 2 for itcovalpc 49509: induction step. (Contributed by AV, 4-May-2024.)

Hypothesis

Ref	Expression
itcovalpc.f	⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))

Assertion

Ref	Expression
itcovalpclem2	⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))

Distinct variable groups:   𝐶,𝑛   𝑦,𝑛

Allowed substitution hints:   𝐶(𝑦)   𝐹(𝑦, 𝑛)

Proof of Theorem itcovalpclem2

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		itcovalpc.f	. . . . 5 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶))
2		nn0ex 12525	. . . . . 6 ⊢ ℕ0 ∈ V
3	2	mptex 7228	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) ∈ V
4	1, 3	eqeltri 2861	. . . 4 ⊢ 𝐹 ∈ V
5		simpl 488	. . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝑦 ∈ ℕ0)
6		simpr 490	. . . 4 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
7		itcovalsucov 49505	. . . 4 ⊢ ((𝐹 ∈ V ∧ 𝑦 ∈ ℕ0 ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
8	4, 5, 6, 7	mp3an2ani 1497	. . 3 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))))
9		simpr 490	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
10		simplr 781	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ ℕ0)
11	5	adantr 486	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑦 ∈ ℕ0)
12	10, 11	nn0mulcld 12585	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℕ0)
13	9, 12	nn0addcld 12584	. . . . . 6 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + (𝐶 · 𝑦)) ∈ ℕ0)
14		eqidd 2766	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
15		oveq1 7426	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 + 𝐶) = (𝑚 + 𝐶))
16	15	cbvmptv 5217	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
17	1, 16	eqtri 2788	. . . . . . 7 ⊢ 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
18	17	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶)))
19		oveq1 7426	. . . . . 6 ⊢ (𝑚 = (𝑛 + (𝐶 · 𝑦)) → (𝑚 + 𝐶) = ((𝑛 + (𝐶 · 𝑦)) + 𝐶))
20	13, 14, 18, 19	fmptco 7129	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)))
21	9	nn0cnd 12582	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
22	12	nn0cnd 12582	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℂ)
23	10	nn0cnd 12582	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ ℂ)
24	21, 22, 23	addassd 11246	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + ((𝐶 · 𝑦) + 𝐶)))
25		nn0cn 12529	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℂ)
26	25	mulridd 11241	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ℕ0 → (𝐶 · 1) = 𝐶)
27	26	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐶 · 1) = 𝐶)
28	27	eqcomd 2771	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐶 = (𝐶 · 1))
29	28	oveq2d 7435	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((𝐶 · 𝑦) + 𝐶) = ((𝐶 · 𝑦) + (𝐶 · 1)))
30		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℕ0)
31	30	nn0cnd 12582	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℂ)
32	5	nn0cnd 12582	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝑦 ∈ ℂ)
33		1cnd 11217	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 1 ∈ ℂ)
34	31, 32, 33	adddid 11248	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐶 · (𝑦 + 1)) = ((𝐶 · 𝑦) + (𝐶 · 1)))
35	29, 34	eqtr4d 2803	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((𝐶 · 𝑦) + 𝐶) = (𝐶 · (𝑦 + 1)))
36	35	oveq2d 7435	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝑛 + ((𝐶 · 𝑦) + 𝐶)) = (𝑛 + (𝐶 · (𝑦 + 1))))
37	36	adantr 486	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + ((𝐶 · 𝑦) + 𝐶)) = (𝑛 + (𝐶 · (𝑦 + 1))))
38	24, 37	eqtrd 2800	. . . . . 6 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + (𝐶 · (𝑦 + 1))))
39	38	mpteq2dva 5206	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
40	20, 39	eqtrd 2800	. . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
41	40	adantr 486	. . 3 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
42	8, 41	eqtrd 2800	. 2 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
43	42	ex 418	1 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ↦ cmpt 5194   ∘ ccom 5667  ‘cfv 6540  (class class class)co 7419  1c1 11116   + caddc 11118   · cmul 11120  ℕ₀cn0 12519  IterCompcitco 49494

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-inf2 9617  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-n0 12520  df-z 12607  df-uz 12879  df-seq 14056  df-itco 49496

This theorem is used by:  itcovalpc  49509