Theorem osumlem5 5534

Description: Lemma for osum 5538.

Hypotheses

Ref	Expression
osumlem5.1	⊢ A ∈ Cℋ
osumlem5.2	⊢ B ∈ Cℋ

Assertion

Ref	Expression
osumlem5	⊢ (H:ℕ–→(A +ℋ B) → ∃f∃g((f:ℕ–→A ∧ g:ℕ–→B) ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w))))

Distinct variable group(s):   w,f,g,A   w,B,f,g   w,H,f,g

Proof of Theorem osumlem5

Step	Hyp	Ref	Expression
1		ffvrn 2890	. . . . 5 ⊢ ((H:ℕ–→(A +ℋ B) ∧ w ∈ ℕ) → (H ‘w) ∈ (A +ℋ B))
2	1	exp 291	. . . 4 ⊢ (H:ℕ–→(A +ℋ B) → (w ∈ ℕ → (H ‘w) ∈ (A +ℋ B)))
3		osumlem5.1	. . . . . 6 ⊢ A ∈ Cℋ
4	3	chshi 5132	. . . . 5 ⊢ A ∈ Sℋ
5		osumlem5.2	. . . . . 6 ⊢ B ∈ Cℋ
6	5	chshi 5132	. . . . 5 ⊢ B ∈ Sℋ
7	4, 6	shsel 5281	. . . 4 ⊢ ((H ‘w) ∈ (A +ℋ B) ↔ ∃x ∈ A ∃y ∈ B (H ‘w) = (x +v y))
8	2, 7	syl6ib 185	. . 3 ⊢ (H:ℕ–→(A +ℋ B) → (w ∈ ℕ → ∃x ∈ A ∃y ∈ B (H ‘w) = (x +v y)))
9	8	r19.21aiv 1259	. 2 ⊢ (H:ℕ–→(A +ℋ B) → ∀w ∈ ℕ ∃x ∈ A ∃y ∈ B (H ‘w) = (x +v y))
10		nnex 4431	. . 3 ⊢ ℕ ∈ V
11	3	elisseti 1355	. . 3 ⊢ A ∈ V
12		opreq1 3006	. . . . 5 ⊢ (x = (f ‘w) → (x +v y) = ((f ‘w) +v y))
13	12	cleq2d 1112	. . . 4 ⊢ (x = (f ‘w) → ((H ‘w) = (x +v y) ↔ (H ‘w) = ((f ‘w) +v y)))
14	13	birexdv 1220	. . 3 ⊢ (x = (f ‘w) → (∃y ∈ B (H ‘w) = (x +v y) ↔ ∃y ∈ B (H ‘w) = ((f ‘w) +v y)))
15	10, 11, 14	ac6 3576	. 2 ⊢ (∀w ∈ ℕ ∃x ∈ A ∃y ∈ B (H ‘w) = (x +v y) → ∃f(f:ℕ–→A ∧ ∀w ∈ ℕ ∃y ∈ B (H ‘w) = ((f ‘w) +v y)))
16	5	elisseti 1355	. . . . . 6 ⊢ B ∈ V
17		opreq2 3007	. . . . . . 7 ⊢ (y = (g ‘w) → ((f ‘w) +v y) = ((f ‘w) +v (g ‘w)))
18	17	cleq2d 1112	. . . . . 6 ⊢ (y = (g ‘w) → ((H ‘w) = ((f ‘w) +v y) ↔ (H ‘w) = ((f ‘w) +v (g ‘w))))
19	10, 16, 18	ac6 3576	. . . . 5 ⊢ (∀w ∈ ℕ ∃y ∈ B (H ‘w) = ((f ‘w) +v y) → ∃g(g:ℕ–→B ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w))))
20	19	anim2i 270	. . . 4 ⊢ ((f:ℕ–→A ∧ ∀w ∈ ℕ ∃y ∈ B (H ‘w) = ((f ‘w) +v y)) → (f:ℕ–→A ∧ ∃g(g:ℕ–→B ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w)))))
21		anass 336	. . . . . 6 ⊢ (((f:ℕ–→A ∧ g:ℕ–→B) ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w))) ↔ (f:ℕ–→A ∧ (g:ℕ–→B ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w)))))
22	21	biex 733	. . . . 5 ⊢ (∃g((f:ℕ–→A ∧ g:ℕ–→B) ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w))) ↔ ∃g(f:ℕ–→A ∧ (g:ℕ–→B ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w)))))
23		19.42v 966	. . . . 5 ⊢ (∃g(f:ℕ–→A ∧ (g:ℕ–→B ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w)))) ↔ (f:ℕ–→A ∧ ∃g(g:ℕ–→B ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w)))))
24	22, 23	bitr 151	. . . 4 ⊢ (∃g((f:ℕ–→A ∧ g:ℕ–→B) ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w))) ↔ (f:ℕ–→A ∧ ∃g(g:ℕ–→B ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w)))))
25	20, 24	sylibr 175	. . 3 ⊢ ((f:ℕ–→A ∧ ∀w ∈ ℕ ∃y ∈ B (H ‘w) = ((f ‘w) +v y)) → ∃g((f:ℕ–→A ∧ g:ℕ–→B) ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w))))
26	25	19.22i 723	. 2 ⊢ (∃f(f:ℕ–→A ∧ ∀w ∈ ℕ ∃y ∈ B (H ‘w) = ((f ‘w) +v y)) → ∃f∃g((f:ℕ–→A ∧ g:ℕ–→B) ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w))))
27	9, 15, 26	3syl 21	1 ⊢ (H:ℕ–→(A +ℋ B) → ∃f∃g((f:ℕ–→A ∧ g:ℕ–→B) ∧ ∀w ∈ ℕ (H ‘w) = ((f ‘w) +v (g ‘w))))

Syntax hints:   → wi 2   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  –→wf 2418   ‘cfv 2422  (class class class)co 3001  ℕcn 4093   +_v cva 4959   C_ℋ cch 4968   +_ℋ cph 4970

This theorem is referenced by:  osumlem6 5535

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080  ax-hilex 4983  ax-hvaddcl 4984

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1o 3104  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-ltp 3884  df-plpr 3958  df-enr 3960  df-nr 3961  df-plr 3962  df-0r 3965  df-1r 3966  df-c 4034  df-1 4036  df-r 4038  df-plus 4039  df-n 4423  df-sh 5114  df-ch 5127  df-shsum 5275

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