Theorem fngsum 13264

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 13135	. 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 3441	. . . 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 3640	. . . . . . 7 ⊢ {(0g‘𝑤)} = {𝑥 ∣ 𝑥 = (0g‘𝑤)}
4		fn0g 13251	. . . . . . . . 9 ⊢ 0g Fn V
5		vex 2776	. . . . . . . . 9 ⊢ 𝑤 ∈ V
6		funfvex 5600	. . . . . . . . . 10 ⊢ ((Fun 0g ∧ 𝑤 ∈ dom 0g) → (0g‘𝑤) ∈ V)
7	6	funfni 5381	. . . . . . . . 9 ⊢ ((0g Fn V ∧ 𝑤 ∈ V) → (0g‘𝑤) ∈ V)
8	4, 5, 7	mp2an 426	. . . . . . . 8 ⊢ (0g‘𝑤) ∈ V
9	8	snex 4233	. . . . . . 7 ⊢ {(0g‘𝑤)} ∈ V
10	3, 9	eqeltrri 2280	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (0g‘𝑤)} ∈ V
11		simpr 110	. . . . . . 7 ⊢ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) → 𝑥 = (0g‘𝑤))
12	11	ss2abi 3266	. . . . . 6 ⊢ {𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ⊆ {𝑥 ∣ 𝑥 = (0g‘𝑤)}
13	10, 12	ssexi 4186	. . . . 5 ⊢ {𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ∈ V
14		zex 9388	. . . . . . 7 ⊢ ℤ ∈ V
15	14, 14	ab2rexex 6223	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)} ∈ V
16		df-rex 2491	. . . . . . . . . . . 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 9660	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ ℤ)
18		eluzelz 9664	. . . . . . . . . . . . . . . 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 1624	. . . . . . . . . . . 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 1931	. . . . . . . . . . 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 2491	. . . . . . . . . . 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 1624	. . . . . . . 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 2491	. . . . . . . 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 3266	. . . . . 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 4186	. . . . 5 ⊢ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))} ∈ V
36	13, 35	unex 4492	. . . 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 2280	. . 3 ⊢ {𝑥 ∣ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))} ∈ V
38		iotaexab 5255	. . 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 6296	1 ⊢ Σg Fn (V × V)

Syntax hints:   ∧ wa 104   ∨ wo 710   = wceq 1373  ∃wex 1516   ∈ wcel 2177  {cab 2192  ∃wrex 2486  Vcvv 2773   ∪ cun 3165  ∅c0 3461  {csn 3634   × cxp 4677  dom cdm 4679  ℩cio 5235   Fn wfn 5271  ‘cfv 5276  (class class class)co 5951  ℤcz 9379  ℤ_≥cuz 9655  ...cfz 10137  seqcseq 10599  +_gcplusg 12953  0_gc0g 13132   Σ_g cgsu 13133

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-neg 8253  df-inn 9044  df-z 9380  df-uz 9656  df-ndx 12879  df-slot 12880  df-base 12882  df-0g 13134  df-igsum 13135

This theorem is referenced by: (None)