Theorem fngsum 13407

Description: Iterated sum has a universal domain. (Contributed by Jim Kingdon, 28-Jun-2025.)

Assertion

Ref	Expression
fngsum	⊢ Σg Fn (V × V)

Proof of Theorem fngsum

Dummy variables 𝑓 𝑚 𝑛 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-igsum 13278	. 2 ⊢ Σg = (𝑤 ∈ V, 𝑓 ∈ V ↦ (℩𝑥((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))))
2		unab 3471	. . . 4 ⊢ ({𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ∪ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))}) = {𝑥 ∣ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))}
3		df-sn 3672	. . . . . . 7 ⊢ {(0g‘𝑤)} = {𝑥 ∣ 𝑥 = (0g‘𝑤)}
4		fn0g 13394	. . . . . . . . 9 ⊢ 0g Fn V
5		vex 2802	. . . . . . . . 9 ⊢ 𝑤 ∈ V
6		funfvex 5640	. . . . . . . . . 10 ⊢ ((Fun 0g ∧ 𝑤 ∈ dom 0g) → (0g‘𝑤) ∈ V)
7	6	funfni 5419	. . . . . . . . 9 ⊢ ((0g Fn V ∧ 𝑤 ∈ V) → (0g‘𝑤) ∈ V)
8	4, 5, 7	mp2an 426	. . . . . . . 8 ⊢ (0g‘𝑤) ∈ V
9	8	snex 4268	. . . . . . 7 ⊢ {(0g‘𝑤)} ∈ V
10	3, 9	eqeltrri 2303	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (0g‘𝑤)} ∈ V
11		simpr 110	. . . . . . 7 ⊢ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) → 𝑥 = (0g‘𝑤))
12	11	ss2abi 3296	. . . . . 6 ⊢ {𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ⊆ {𝑥 ∣ 𝑥 = (0g‘𝑤)}
13	10, 12	ssexi 4221	. . . . 5 ⊢ {𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ∈ V
14		zex 9443	. . . . . . 7 ⊢ ℤ ∈ V
15	14, 14	ab2rexex 6266	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)} ∈ V
16		df-rex 2514	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)) ↔ ∃𝑛(𝑛 ∈ (ℤ≥‘𝑚) ∧ (dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))))
17		eluzel2 9715	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ ℤ)
18		eluzelz 9719	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑛 ∈ ℤ)
19	17, 18	jca 306	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ))
20		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)) → 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))
21	19, 20	anim12i 338	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ (dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))) → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))
22		anass 401	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)) ↔ (𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))))
23	21, 22	sylib 122	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ (ℤ≥‘𝑚) ∧ (dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))) → (𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))))
24	23	eximi 1646	. . . . . . . . . . . 12 ⊢ (∃𝑛(𝑛 ∈ (ℤ≥‘𝑚) ∧ (dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))) → ∃𝑛(𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))))
25	16, 24	sylbi 121	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)) → ∃𝑛(𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))))
26		19.42v 1953	. . . . . . . . . . 11 ⊢ (∃𝑛(𝑚 ∈ ℤ ∧ (𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))) ↔ (𝑚 ∈ ℤ ∧ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))))
27	25, 26	sylib 122	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)) → (𝑚 ∈ ℤ ∧ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))))
28		df-rex 2514	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛) ↔ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))
29	28	anbi2i 457	. . . . . . . . . 10 ⊢ ((𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)) ↔ (𝑚 ∈ ℤ ∧ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))))
30	27, 29	sylibr 134	. . . . . . . . 9 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)) → (𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))
31	30	eximi 1646	. . . . . . . 8 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)) → ∃𝑚(𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))
32		df-rex 2514	. . . . . . . 8 ⊢ (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛) ↔ ∃𝑚(𝑚 ∈ ℤ ∧ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))
33	31, 32	sylibr 134	. . . . . . 7 ⊢ (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)) → ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))
34	33	ss2abi 3296	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))} ⊆ {𝑥 ∣ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)}
35	15, 34	ssexi 4221	. . . . 5 ⊢ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))} ∈ V
36	13, 35	unex 4529	. . . 4 ⊢ ({𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ∪ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))}) ∈ V
37	2, 36	eqeltrri 2303	. . 3 ⊢ {𝑥 ∣ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))} ∈ V
38		iotaexab 5293	. . 3 ⊢ ({𝑥 ∣ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))} ∈ V → (℩𝑥((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))) ∈ V)
39	37, 38	ax-mp 5	. 2 ⊢ (℩𝑥((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))) ∈ V
40	1, 39	fnmpoi 6339	1 ⊢ Σg Fn (V × V)

Syntax hints:   ∧ wa 104   ∨ wo 713   = wceq 1395  ∃wex 1538   ∈ wcel 2200  {cab 2215  ∃wrex 2509  Vcvv 2799   ∪ cun 3195  ∅c0 3491  {csn 3666   × cxp 4714  dom cdm 4716  ℩cio 5272   Fn wfn 5309  ‘cfv 5314  (class class class)co 5994  ℤcz 9434  ℤ_≥cuz 9710  ...cfz 10192  seqcseq 10656  +_gcplusg 13096  0_gc0g 13275   Σ_g cgsu 13276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-neg 8308  df-inn 9099  df-z 9435  df-uz 9711  df-ndx 13021  df-slot 13022  df-base 13024  df-0g 13277  df-igsum 13278

This theorem is referenced by: (None)