Theorem itcovalpclem2 48651

Description: Lemma 2 for itcovalpc 48652: 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 12507	. . . . . 6 ⊢ ℕ0 ∈ V
3	2	mptex 7215	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) ∈ V
4	1, 3	eqeltri 2830	. . . 4 ⊢ 𝐹 ∈ V
5		simpl 482	. . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝑦 ∈ ℕ0)
6		simpr 484	. . . 4 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) → ((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
7		itcovalsucov 48648	. . . 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 1470	. . 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 484	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
10		simplr 768	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ ℕ0)
11	5	adantr 480	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑦 ∈ ℕ0)
12	10, 11	nn0mulcld 12567	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℕ0)
13	9, 12	nn0addcld 12566	. . . . . 6 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + (𝐶 · 𝑦)) ∈ ℕ0)
14		eqidd 2736	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
15		oveq1 7412	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 + 𝐶) = (𝑚 + 𝐶))
16	15	cbvmptv 5225	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
17	1, 16	eqtri 2758	. . . . . . 7 ⊢ 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
18	17	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶)))
19		oveq1 7412	. . . . . 6 ⊢ (𝑚 = (𝑛 + (𝐶 · 𝑦)) → (𝑚 + 𝐶) = ((𝑛 + (𝐶 · 𝑦)) + 𝐶))
20	13, 14, 18, 19	fmptco 7119	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)))
21	9	nn0cnd 12564	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
22	12	nn0cnd 12564	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℂ)
23	10	nn0cnd 12564	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ ℂ)
24	21, 22, 23	addassd 11257	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + ((𝐶 · 𝑦) + 𝐶)))
25		nn0cn 12511	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℂ)
26	25	mulridd 11252	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ℕ0 → (𝐶 · 1) = 𝐶)
27	26	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐶 · 1) = 𝐶)
28	27	eqcomd 2741	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐶 = (𝐶 · 1))
29	28	oveq2d 7421	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((𝐶 · 𝑦) + 𝐶) = ((𝐶 · 𝑦) + (𝐶 · 1)))
30		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℕ0)
31	30	nn0cnd 12564	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℂ)
32	5	nn0cnd 12564	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝑦 ∈ ℂ)
33		1cnd 11230	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 1 ∈ ℂ)
34	31, 32, 33	adddid 11259	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐶 · (𝑦 + 1)) = ((𝐶 · 𝑦) + (𝐶 · 1)))
35	29, 34	eqtr4d 2773	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((𝐶 · 𝑦) + 𝐶) = (𝐶 · (𝑦 + 1)))
36	35	oveq2d 7421	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝑛 + ((𝐶 · 𝑦) + 𝐶)) = (𝑛 + (𝐶 · (𝑦 + 1))))
37	36	adantr 480	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + ((𝐶 · 𝑦) + 𝐶)) = (𝑛 + (𝐶 · (𝑦 + 1))))
38	24, 37	eqtrd 2770	. . . . . 6 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + (𝐶 · (𝑦 + 1))))
39	38	mpteq2dva 5214	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
40	20, 39	eqtrd 2770	. . . 4 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
41	40	adantr 480	. . 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 2770	. 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 412	1 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (((IterComp‘𝐹)‘𝑦) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) → ((IterComp‘𝐹)‘(𝑦 + 1)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1))))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Vcvv 3459   ↦ cmpt 5201   ∘ ccom 5658  ‘cfv 6531  (class class class)co 7405  1c1 11130   + caddc 11132   · cmul 11134  ℕ₀cn0 12501  IterCompcitco 48637

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-n0 12502  df-z 12589  df-uz 12853  df-seq 14020  df-itco 48639

This theorem is referenced by:  itcovalpc  48652