Theorem gsumval3a 19016

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 2798	. . 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	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8		gsumval3.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
9	7, 8	jca 515	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
10		fex 6966	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
12	2	fvexi 6659	. . . . 5 ⊢ 0 ∈ V
13		suppimacnv 7824	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
14	11, 12, 13	sylancl 589	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
15		gsumval3.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
16	1, 2, 3, 4	gsumvallem2 17990	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
17	15, 16	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
18	17	eqcomd 2804	. . . . . 6 ⊢ (𝜑 → { 0 } = {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
19	18	difeq2d 4050	. . . . 5 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
20	19	imaeq2d 5896	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
21	6, 14, 20	3eqtrd 2837	. . 3 ⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
22	1, 2, 3, 4, 21, 15, 8, 7	gsumval 17879	. 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)))))))
23		gsumval3a.n	. . . 4 ⊢ (𝜑 → 𝑊 ≠ ∅)
24	17	sseq2d 3947	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} ↔ ran 𝐹 ⊆ { 0 }))
25	5	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = (𝐹 supp 0 ))
26	9	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
27	26, 10	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V)
28	27, 12, 13	sylancl 589	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
29	7	ffnd 6488	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝐴)
30	29	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴)
31		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → ran 𝐹 ⊆ { 0 })
32		df-f 6328	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 }))
33	30, 31, 32	sylanbrc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 })
34		disjdif 4379	. . . . . . . . 9 ⊢ ({ 0 } ∩ (V ∖ { 0 })) = ∅
35		fimacnvdisj 6531	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V ∖ { 0 })) = ∅) → (◡𝐹 “ (V ∖ { 0 })) = ∅)
36	33, 34, 35	sylancl 589	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (◡𝐹 “ (V ∖ { 0 })) = ∅)
37	25, 28, 36	3eqtrd 2837	. . . . . . 7 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅)
38	37	ex 416	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅))
39	24, 38	sylbid 243	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅))
40	39	necon3ad 3000	. . . 4 ⊢ (𝜑 → (𝑊 ≠ ∅ → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
41	23, 40	mpd 15	. . 3 ⊢ (𝜑 → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
42	41	iffalsed 4436	. 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 4436	. 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	22, 42, 44	3eqtrd 2837	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 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878   ∩ cin 3880   ⊆ wss 3881  ∅c0 4243  ifcif 4425  {csn 4525  ^◡ccnv 5518  ran crn 5520   “ cima 5522   ∘ ccom 5523  ℩cio 6281   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135   supp csupp 7813  Fincfn 8492  1c1 10527  ℤ_≥cuz 12231  ...cfz 12885  seqcseq 13364  ♯chash 13686  Basecbs 16475  +_gcplusg 16557  0_gc0g 16705   Σ_g cgsu 16706  Mndcmnd 17903  Cntzccntz 18437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-supp 7814  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-seq 13365  df-0g 16707  df-gsum 16708  df-mgm 17844  df-sgrp 17893  df-mnd 17904

This theorem is referenced by:  gsumval3lem2  19019