Theorem climaddf 41986

Description: A version of climadd 14973 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
climaddf.1	⊢ Ⅎ𝑘𝜑
climaddf.2	⊢ Ⅎ𝑘𝐹
climaddf.3	⊢ Ⅎ𝑘𝐺
climaddf.4	⊢ Ⅎ𝑘𝐻
climaddf.5	⊢ 𝑍 = (ℤ≥‘𝑀)
climaddf.6	⊢ (𝜑 → 𝑀 ∈ ℤ)
climaddf.7	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climaddf.8	⊢ (𝜑 → 𝐻 ∈ 𝑋)
climaddf.9	⊢ (𝜑 → 𝐺 ⇝ 𝐵)
climaddf.10	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
climaddf.11	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
climaddf.12	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))

Assertion

Ref	Expression
climaddf	⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵))

Distinct variable group:   𝑘,𝑍

Allowed substitution hints:   𝜑(𝑘)   𝐴(𝑘)   𝐵(𝑘)   𝐹(𝑘)   𝐺(𝑘)   𝐻(𝑘)   𝑀(𝑘)   𝑋(𝑘)

Proof of Theorem climaddf

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		climaddf.5	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
2		climaddf.6	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		climaddf.7	. 2 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
4		climaddf.8	. 2 ⊢ (𝜑 → 𝐻 ∈ 𝑋)
5		climaddf.9	. 2 ⊢ (𝜑 → 𝐺 ⇝ 𝐵)
6		climaddf.1	. . . . 5 ⊢ Ⅎ𝑘𝜑
7		nfv 1915	. . . . 5 ⊢ Ⅎ𝑘 𝑗 ∈ 𝑍
8	6, 7	nfan 1900	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ 𝑍)
9		climaddf.2	. . . . . 6 ⊢ Ⅎ𝑘𝐹
10		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑘𝑗
11	9, 10	nffv 6666	. . . . 5 ⊢ Ⅎ𝑘(𝐹‘𝑗)
12	11	nfel1 2994	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑗) ∈ ℂ
13	8, 12	nfim 1897	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
14		eleq1w 2895	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝑘 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
15	14	anbi2d 630	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝜑 ∧ 𝑘 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
16		fveq2 6656	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐹‘𝑘) = (𝐹‘𝑗))
17	16	eleq1d 2897	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘𝑗) ∈ ℂ))
18	15, 17	imbi12d 347	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)))
19		climaddf.10	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ)
20	13, 18, 19	chvarfv 2242	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) ∈ ℂ)
21		climaddf.3	. . . . . 6 ⊢ Ⅎ𝑘𝐺
22	21, 10	nffv 6666	. . . . 5 ⊢ Ⅎ𝑘(𝐺‘𝑗)
23	22	nfel1 2994	. . . 4 ⊢ Ⅎ𝑘(𝐺‘𝑗) ∈ ℂ
24	8, 23	nfim 1897	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ)
25		fveq2 6656	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐺‘𝑘) = (𝐺‘𝑗))
26	25	eleq1d 2897	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐺‘𝑘) ∈ ℂ ↔ (𝐺‘𝑗) ∈ ℂ))
27	15, 26	imbi12d 347	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ)))
28		climaddf.11	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐺‘𝑘) ∈ ℂ)
29	24, 27, 28	chvarfv 2242	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐺‘𝑗) ∈ ℂ)
30		climaddf.4	. . . . . 6 ⊢ Ⅎ𝑘𝐻
31	30, 10	nffv 6666	. . . . 5 ⊢ Ⅎ𝑘(𝐻‘𝑗)
32		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑘 +
33	11, 32, 22	nfov 7172	. . . . 5 ⊢ Ⅎ𝑘((𝐹‘𝑗) + (𝐺‘𝑗))
34	31, 33	nfeq 2991	. . . 4 ⊢ Ⅎ𝑘(𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗))
35	8, 34	nfim 1897	. . 3 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗)))
36		fveq2 6656	. . . . 5 ⊢ (𝑘 = 𝑗 → (𝐻‘𝑘) = (𝐻‘𝑗))
37	16, 25	oveq12d 7160	. . . . 5 ⊢ (𝑘 = 𝑗 → ((𝐹‘𝑘) + (𝐺‘𝑘)) = ((𝐹‘𝑗) + (𝐺‘𝑗)))
38	36, 37	eqeq12d 2837	. . . 4 ⊢ (𝑘 = 𝑗 → ((𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)) ↔ (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗))))
39	15, 38	imbi12d 347	. . 3 ⊢ (𝑘 = 𝑗 → (((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘))) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗)))))
40		climaddf.12	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐻‘𝑘) = ((𝐹‘𝑘) + (𝐺‘𝑘)))
41	35, 39, 40	chvarfv 2242	. 2 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐻‘𝑗) = ((𝐹‘𝑗) + (𝐺‘𝑗)))
42	1, 2, 3, 4, 5, 20, 29, 41	climadd 14973	1 ⊢ (𝜑 → 𝐻 ⇝ (𝐴 + 𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  Ⅎwnfc 2961   class class class wbr 5052  ‘cfv 6341  (class class class)co 7142  ℂcc 10521   + caddc 10526  ℤcz 11968  ℤ_≥cuz 12230   ⇝ cli 14826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5189  ax-nul 5196  ax-pow 5252  ax-pr 5316  ax-un 7447  ax-cnex 10579  ax-resscn 10580  ax-1cn 10581  ax-icn 10582  ax-addcl 10583  ax-addrcl 10584  ax-mulcl 10585  ax-mulrcl 10586  ax-mulcom 10587  ax-addass 10588  ax-mulass 10589  ax-distr 10590  ax-i2m1 10591  ax-1ne0 10592  ax-1rid 10593  ax-rnegex 10594  ax-rrecex 10595  ax-cnre 10596  ax-pre-lttri 10597  ax-pre-lttrn 10598  ax-pre-ltadd 10599  ax-pre-mulgt0 10600  ax-pre-sup 10601

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3488  df-sbc 3764  df-csb 3872  df-dif 3927  df-un 3929  df-in 3931  df-ss 3940  df-pss 3942  df-nul 4280  df-if 4454  df-pw 4527  df-sn 4554  df-pr 4556  df-tp 4558  df-op 4560  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5446  df-eprel 5451  df-po 5460  df-so 5461  df-fr 5500  df-we 5502  df-xp 5547  df-rel 5548  df-cnv 5549  df-co 5550  df-dm 5551  df-rn 5552  df-res 5553  df-ima 5554  df-pred 6134  df-ord 6180  df-on 6181  df-lim 6182  df-suc 6183  df-iota 6300  df-fun 6343  df-fn 6344  df-f 6345  df-f1 6346  df-fo 6347  df-f1o 6348  df-fv 6349  df-riota 7100  df-ov 7145  df-oprab 7146  df-mpo 7147  df-om 7567  df-2nd 7676  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-er 8275  df-en 8496  df-dom 8497  df-sdom 8498  df-sup 8892  df-pnf 10663  df-mnf 10664  df-xr 10665  df-ltxr 10666  df-le 10667  df-sub 10858  df-neg 10859  df-div 11284  df-nn 11625  df-2 11687  df-3 11688  df-n0 11885  df-z 11969  df-uz 12231  df-rp 12377  df-seq 13360  df-exp 13420  df-cj 14443  df-re 14444  df-im 14445  df-sqrt 14579  df-abs 14580  df-clim 14830

This theorem is referenced by:  fourierdlem112  42593