Theorem gsumval3a 19968

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 2763	. . 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 7225	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
10	2	fvexi 6895	. . . . 5 ⊢ 0 ∈ V
11		suppimacnv 8166	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
12	9, 10, 11	sylancl 597	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
13		gsumval3.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
14	1, 2, 3, 4	gsumvallem2 18888	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
15	13, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
16	15	eqcomd 2769	. . . . . 6 ⊢ (𝜑 → { 0 } = {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
17	16	difeq2d 4081	. . . . 5 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
18	17	imaeq2d 6062	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
19	6, 12, 18	3eqtrd 2802	. . 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 18730	. 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 3969	. . . . . 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 520	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
25	24	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
26		fex 7224	. . . . . . . . . 10 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
27	25, 26	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V)
28	27, 10, 11	sylancl 597	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
29	7	ffnd 6706	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝐴)
30	29	adantr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴)
31		simpr 489	. . . . . . . . . 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 594	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 })
34		disjdif 4433	. . . . . . . . 9 ⊢ ({ 0 } ∩ (V ∖ { 0 })) = ∅
35		fimacnvdisj 6756	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V ∖ { 0 })) = ∅) → (◡𝐹 “ (V ∖ { 0 })) = ∅)
36	33, 34, 35	sylancl 597	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (◡𝐹 “ (V ∖ { 0 })) = ∅)
37	23, 28, 36	3eqtrd 2802	. . . . . . 7 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅)
38	37	ex 417	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅))
39	22, 38	sylbid 243	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅))
40	39	necon3ad 2971	. . . 4 ⊢ (𝜑 → (𝑊 ≠ ∅ → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
41	21, 40	mpd 16	. . 3 ⊢ (𝜑 → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
42	41	iffalsed 4498	. 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 4498	. 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 2802	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 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  ifcif 4487  {csn 4589  ^◡ccnv 5660  ran crn 5662   “ cima 5664   ∘ ccom 5665  ℩cio 6490   Fn wfn 6531  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410   supp csupp 8152  Fincfn 8939  1c1 11096  ℤ_≥cuz 12857  ...cfz 13530  seqcseq 14033  ♯chash 14362  Basecbs 17264  +_gcplusg 17305  0_gc0g 17487   Σ_g cgsu 17488  Mndcmnd 18787  Cntzccntz 19380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-iota 6492  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 8153  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-seq 14034  df-0g 17489  df-gsum 17490  df-mgm 18693  df-sgrp 18772  df-mnd 18788

This theorem is referenced by:  gsumval3lem2  19971