Theorem fngsum 13532

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 13403	. 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 3476	. . . 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 3679	. . . . . . 7 ⊢ {(0g‘𝑤)} = {𝑥 ∣ 𝑥 = (0g‘𝑤)}
4		fn0g 13519	. . . . . . . . 9 ⊢ 0g Fn V
5		vex 2806	. . . . . . . . 9 ⊢ 𝑤 ∈ V
6		funfvex 5665	. . . . . . . . . 10 ⊢ ((Fun 0g ∧ 𝑤 ∈ dom 0g) → (0g‘𝑤) ∈ V)
7	6	funfni 5439	. . . . . . . . 9 ⊢ ((0g Fn V ∧ 𝑤 ∈ V) → (0g‘𝑤) ∈ V)
8	4, 5, 7	mp2an 426	. . . . . . . 8 ⊢ (0g‘𝑤) ∈ V
9	8	snex 4281	. . . . . . 7 ⊢ {(0g‘𝑤)} ∈ V
10	3, 9	eqeltrri 2305	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (0g‘𝑤)} ∈ V
11		simpr 110	. . . . . . 7 ⊢ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) → 𝑥 = (0g‘𝑤))
12	11	ss2abi 3300	. . . . . 6 ⊢ {𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ⊆ {𝑥 ∣ 𝑥 = (0g‘𝑤)}
13	10, 12	ssexi 4232	. . . . 5 ⊢ {𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ∈ V
14		zex 9531	. . . . . . 7 ⊢ ℤ ∈ V
15	14, 14	ab2rexex 6302	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)} ∈ V
16		df-rex 2517	. . . . . . . . . . . 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 9803	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ ℤ)
18		eluzelz 9808	. . . . . . . . . . . . . . . 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 1649	. . . . . . . . . . . 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 1955	. . . . . . . . . . 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 2517	. . . . . . . . . . 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 1649	. . . . . . . 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 2517	. . . . . . . 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 3300	. . . . . 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 4232	. . . . 5 ⊢ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))} ∈ V
36	13, 35	unex 4544	. . . 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 2305	. . 3 ⊢ {𝑥 ∣ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))} ∈ V
38		iotaexab 5312	. . 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 6377	1 ⊢ Σg Fn (V × V)

Syntax hints:   ∧ wa 104   ∨ wo 716   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {cab 2217  ∃wrex 2512  Vcvv 2803   ∪ cun 3199  ∅c0 3496  {csn 3673   × cxp 4729  dom cdm 4731  ℩cio 5291   Fn wfn 5328  ‘cfv 5333  (class class class)co 6028  ℤcz 9522  ℤ_≥cuz 9798  ...cfz 10286  seqcseq 10753  +_gcplusg 13221  0_gc0g 13400   Σ_g cgsu 13401

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-neg 8396  df-inn 9187  df-z 9523  df-uz 9799  df-ndx 13146  df-slot 13147  df-base 13149  df-0g 13402  df-igsum 13403

This theorem is referenced by:  gfsumval  16789