Theorem gsumval3a 19889

Description: Value of the group sum operation over an index set with finite support. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by AV, 29-May-2019.)

Hypotheses

Ref	Expression
gsumval3.b	⊢ 𝐵 = (Base‘𝐺)
gsumval3.0	⊢ 0 = (0g‘𝐺)
gsumval3.p	⊢ + = (+g‘𝐺)
gsumval3.z	⊢ 𝑍 = (Cntz‘𝐺)
gsumval3.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
gsumval3.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
gsumval3.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
gsumval3.c	⊢ (𝜑 → ran 𝐹 ⊆ (𝑍‘ran 𝐹))
gsumval3a.t	⊢ (𝜑 → 𝑊 ∈ Fin)
gsumval3a.n	⊢ (𝜑 → 𝑊 ≠ ∅)
gsumval3a.w	⊢ 𝑊 = (𝐹 supp 0 )
gsumval3a.i	⊢ (𝜑 → ¬ 𝐴 ∈ ran ...)

Assertion

Ref	Expression
gsumval3a	⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))

Distinct variable groups:   𝑥,𝑓, +   𝐴,𝑓,𝑥   𝜑,𝑓,𝑥   𝑥, 0   𝑓,𝐺,𝑥   𝑥,𝑉   𝐵,𝑓,𝑥   𝑓,𝐹,𝑥   𝑓,𝑊,𝑥

Allowed substitution hints:   𝑉(𝑓)   0 (𝑓)   𝑍(𝑥,𝑓)

Proof of Theorem gsumval3a

Dummy variables 𝑚 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gsumval3.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		gsumval3.0	. . 3 ⊢ 0 = (0g‘𝐺)
3		gsumval3.p	. . 3 ⊢ + = (+g‘𝐺)
4		eqid 2736	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}
5		gsumval3a.w	. . . . 5 ⊢ 𝑊 = (𝐹 supp 0 )
6	5	a1i 11	. . . 4 ⊢ (𝜑 → 𝑊 = (𝐹 supp 0 ))
7		gsumval3.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8		gsumval3.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9	7, 8	fexd 7224	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
10	2	fvexi 6895	. . . . 5 ⊢ 0 ∈ V
11		suppimacnv 8178	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
12	9, 10, 11	sylancl 586	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
13		gsumval3.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
14	1, 2, 3, 4	gsumvallem2 18817	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
15	13, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
16	15	eqcomd 2742	. . . . . 6 ⊢ (𝜑 → { 0 } = {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
17	16	difeq2d 4106	. . . . 5 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
18	17	imaeq2d 6052	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
19	6, 12, 18	3eqtrd 2775	. . 3 ⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
20	1, 2, 3, 4, 19, 13, 8, 7	gsumval 18660	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))))
21		gsumval3a.n	. . . 4 ⊢ (𝜑 → 𝑊 ≠ ∅)
22	15	sseq2d 3996	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} ↔ ran 𝐹 ⊆ { 0 }))
23	5	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = (𝐹 supp 0 ))
24	7, 8	jca 511	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
25	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
26		fex 7223	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
27	25, 26	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V)
28	27, 10, 11	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
29	7	ffnd 6712	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝐴)
30	29	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴)
31		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → ran 𝐹 ⊆ { 0 })
32		df-f 6540	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 }))
33	30, 31, 32	sylanbrc 583	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 })
34		disjdif 4452	. . . . . . . . 9 ⊢ ({ 0 } ∩ (V ∖ { 0 })) = ∅
35		fimacnvdisj 6761	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V ∖ { 0 })) = ∅) → (◡𝐹 “ (V ∖ { 0 })) = ∅)
36	33, 34, 35	sylancl 586	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (◡𝐹 “ (V ∖ { 0 })) = ∅)
37	23, 28, 36	3eqtrd 2775	. . . . . . 7 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅)
38	37	ex 412	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅))
39	22, 38	sylbid 240	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅))
40	39	necon3ad 2946	. . . 4 ⊢ (𝜑 → (𝑊 ≠ ∅ → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
41	21, 40	mpd 15	. . 3 ⊢ (𝜑 → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
42	41	iffalsed 4516	. 2 ⊢ (𝜑 → if(ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))) = if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))))
43		gsumval3a.i	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ ran ...)
44	43	iffalsed 4516	. 2 ⊢ (𝜑 → if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))) = (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))
45	20, 42, 44	3eqtrd 2775	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊)))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  {crab 3420  Vcvv 3464   ∖ cdif 3928   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  ifcif 4505  {csn 4606  ^◡ccnv 5658  ran crn 5660   “ cima 5662   ∘ ccom 5663  ℩cio 6487   Fn wfn 6531  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   supp csupp 8164  Fincfn 8964  1c1 11135  ℤ_≥cuz 12857  ...cfz 13529  seqcseq 14024  ♯chash 14353  Basecbs 17233  +_gcplusg 17276  0_gc0g 17458   Σ_g cgsu 17459  Mndcmnd 18717  Cntzccntz 19303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-supp 8165  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-seq 14025  df-0g 17460  df-gsum 17461  df-mgm 18623  df-sgrp 18702  df-mnd 18718

This theorem is referenced by:  gsumval3lem2  19892