Theorem fngzsum 13685

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

Assertion

Ref	Expression
fngzsum	⊢ Σgz Fn (V × V)

Proof of Theorem fngzsum

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

Step	Hyp	Ref	Expression
1		df-gzsum 13590	. 2 ⊢ Σgz = (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 3498	. . . 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 3711	. . . . . . 7 ⊢ {(0g‘𝑤)} = {𝑥 ∣ 𝑥 = (0g‘𝑤)}
4		fn0g 13672	. . . . . . . . 9 ⊢ 0g Fn V
5		vex 2824	. . . . . . . . 9 ⊢ 𝑤 ∈ V
6		funfvex 5707	. . . . . . . . . 10 ⊢ ((Fun 0g ∧ 𝑤 ∈ dom 0g) → (0g‘𝑤) ∈ V)
7	6	funfni 5478	. . . . . . . . 9 ⊢ ((0g Fn V ∧ 𝑤 ∈ V) → (0g‘𝑤) ∈ V)
8	4, 5, 7	mp2an 430	. . . . . . . 8 ⊢ (0g‘𝑤) ∈ V
9	8	snex 4317	. . . . . . 7 ⊢ {(0g‘𝑤)} ∈ V
10	3, 9	eqeltrri 2312	. . . . . 6 ⊢ {𝑥 ∣ 𝑥 = (0g‘𝑤)} ∈ V
11		simpr 110	. . . . . . 7 ⊢ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) → 𝑥 = (0g‘𝑤))
12	11	ss2abi 3320	. . . . . 6 ⊢ {𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ⊆ {𝑥 ∣ 𝑥 = (0g‘𝑤)}
13	10, 12	ssexi 4266	. . . . 5 ⊢ {𝑥 ∣ (dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤))} ∈ V
14		zex 9632	. . . . . . 7 ⊢ ℤ ∈ V
15	14, 14	ab2rexex 6354	. . . . . 6 ⊢ {𝑥 ∣ ∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)} ∈ V
16		df-rex 2534	. . . . . . . . . . . 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 9905	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑚 ∈ ℤ)
18		eluzelz 9910	. . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . 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 1653	. . . . . . . . . . . 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 1962	. . . . . . . . . . 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 2534	. . . . . . . . . . 11 ⊢ (∃𝑛 ∈ ℤ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛) ↔ ∃𝑛(𝑛 ∈ ℤ ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))
29	28	anbi2i 461	. . . . . . . . . 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 1653	. . . . . . . 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 2534	. . . . . . . 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 3320	. . . . . 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 4266	. . . . 5 ⊢ {𝑥 ∣ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛))} ∈ V
36	13, 35	unex 4582	. . . 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 2312	. . 3 ⊢ {𝑥 ∣ ((dom 𝑓 = ∅ ∧ 𝑥 = (0g‘𝑤)) ∨ ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(dom 𝑓 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚((+g‘𝑤), 𝑓)‘𝑛)))} ∈ V
38		iotaexab 5351	. . 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 6429	1 ⊢ Σgz Fn (V × V)

Syntax hints:   ∧ wa 104   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∃wrex 2529  Vcvv 2821   ∪ cun 3218  ∅c0 3520  {csn 3705   × cxp 4767  dom cdm 4769  ℩cio 5330   Fn wfn 5367  ‘cfv 5372  (class class class)co 6075  ℤcz 9623  ℤ_≥cuz 9900  ...cfz 10390  seqcseq 10862  +_gcplusg 13408  0_gc0g 13587   Σ_gz cgzsu 13588

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-neg 8490  df-inn 9284  df-z 9624  df-uz 9901  df-ndx 13333  df-slot 13334  df-base 13336  df-0g 13589  df-gzsum 13590

This theorem is referenced by:  gsumvalfi  14129