Theorem itcovalpclem2 48405

Description: Lemma 2 for itcovalpc 48406: 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 12559	. . . . . 6 ⊢ ℕ0 ∈ V
3	2	mptex 7260	. . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) ∈ V
4	1, 3	eqeltri 2840	. . . 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 48402	. . . 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 1468	. . 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 12618	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℕ0)
13	9, 12	nn0addcld 12617	. . . . . 6 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝑛 + (𝐶 · 𝑦)) ∈ ℕ0)
14		eqidd 2741	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦))))
15		oveq1 7455	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → (𝑛 + 𝐶) = (𝑚 + 𝐶))
16	15	cbvmptv 5279	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ0 ↦ (𝑛 + 𝐶)) = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
17	1, 16	eqtri 2768	. . . . . . 7 ⊢ 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶))
18	17	a1i 11	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐹 = (𝑚 ∈ ℕ0 ↦ (𝑚 + 𝐶)))
19		oveq1 7455	. . . . . 6 ⊢ (𝑚 = (𝑛 + (𝐶 · 𝑦)) → (𝑚 + 𝐶) = ((𝑛 + (𝐶 · 𝑦)) + 𝐶))
20	13, 14, 18, 19	fmptco 7163	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐹 ∘ (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · 𝑦)))) = (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)))
21	9	nn0cnd 12615	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℂ)
22	12	nn0cnd 12615	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → (𝐶 · 𝑦) ∈ ℂ)
23	10	nn0cnd 12615	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → 𝐶 ∈ ℂ)
24	21, 22, 23	addassd 11312	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + ((𝐶 · 𝑦) + 𝐶)))
25		nn0cn 12563	. . . . . . . . . . . . . 14 ⊢ (𝐶 ∈ ℕ0 → 𝐶 ∈ ℂ)
26	25	mulridd 11307	. . . . . . . . . . . . 13 ⊢ (𝐶 ∈ ℕ0 → (𝐶 · 1) = 𝐶)
27	26	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐶 · 1) = 𝐶)
28	27	eqcomd 2746	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐶 = (𝐶 · 1))
29	28	oveq2d 7464	. . . . . . . . . 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 12615	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝐶 ∈ ℂ)
32	5	nn0cnd 12615	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 𝑦 ∈ ℂ)
33		1cnd 11285	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → 1 ∈ ℂ)
34	31, 32, 33	adddid 11314	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝐶 · (𝑦 + 1)) = ((𝐶 · 𝑦) + (𝐶 · 1)))
35	29, 34	eqtr4d 2783	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → ((𝐶 · 𝑦) + 𝐶) = (𝐶 · (𝑦 + 1)))
36	35	oveq2d 7464	. . . . . . . 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 2780	. . . . . 6 ⊢ (((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) ∧ 𝑛 ∈ ℕ0) → ((𝑛 + (𝐶 · 𝑦)) + 𝐶) = (𝑛 + (𝐶 · (𝑦 + 1))))
39	38	mpteq2dva 5266	. . . . 5 ⊢ ((𝑦 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ ((𝑛 + (𝐶 · 𝑦)) + 𝐶)) = (𝑛 ∈ ℕ0 ↦ (𝑛 + (𝐶 · (𝑦 + 1)))))
40	20, 39	eqtrd 2780	. . . 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 2780	. 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 1537   ∈ wcel 2108  Vcvv 3488   ↦ cmpt 5249   ∘ ccom 5704  ‘cfv 6573  (class class class)co 7448  1c1 11185   + caddc 11187   · cmul 11189  ℕ₀cn0 12553  IterCompcitco 48391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-nn 12294  df-n0 12554  df-z 12640  df-uz 12904  df-seq 14053  df-itco 48393

This theorem is referenced by:  itcovalpc  48406