Theorem gsumval3a 18127

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 2610	. . 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 553	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
10		fex 6394	. . . . . 6 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
11	9, 10	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ V)
12		fvex 6113	. . . . . 6 ⊢ (0g‘𝐺) ∈ V
13	2, 12	eqeltri 2684	. . . . 5 ⊢ 0 ∈ V
14		suppimacnv 7193	. . . . 5 ⊢ ((𝐹 ∈ V ∧ 0 ∈ V) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
15	11, 13, 14	sylancl 693	. . . 4 ⊢ (𝜑 → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
16		gsumval3.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
17	1, 2, 3, 4	gsumvallem2 17195	. . . . . . . 8 ⊢ (𝐺 ∈ Mnd → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} = { 0 })
19	18	eqcomd 2616	. . . . . 6 ⊢ (𝜑 → { 0 } = {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
20	19	difeq2d 3690	. . . . 5 ⊢ (𝜑 → (V ∖ { 0 }) = (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
21	20	imaeq2d 5385	. . . 4 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
22	6, 15, 21	3eqtrd 2648	. . 3 ⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})))
23	1, 2, 3, 4, 22, 16, 8, 7	gsumval 17094	. 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)))))))
24		gsumval3a.n	. . . 4 ⊢ (𝜑 → 𝑊 ≠ ∅)
25	18	sseq2d 3596	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} ↔ ran 𝐹 ⊆ { 0 }))
26	5	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = (𝐹 supp 0 ))
27	9	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉))
28	27, 10	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 ∈ V)
29	28, 13, 14	sylancl 693	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (𝐹 supp 0 ) = (◡𝐹 “ (V ∖ { 0 })))
30		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
31	7, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝐴)
32	31	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹 Fn 𝐴)
33		simpr 476	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → ran 𝐹 ⊆ { 0 })
34		df-f 5808	. . . . . . . . . 10 ⊢ (𝐹:𝐴⟶{ 0 } ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ { 0 }))
35	32, 33, 34	sylanbrc 695	. . . . . . . . 9 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝐹:𝐴⟶{ 0 })
36		disjdif 3992	. . . . . . . . 9 ⊢ ({ 0 } ∩ (V ∖ { 0 })) = ∅
37		fimacnvdisj 5996	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶{ 0 } ∧ ({ 0 } ∩ (V ∖ { 0 })) = ∅) → (◡𝐹 “ (V ∖ { 0 })) = ∅)
38	35, 36, 37	sylancl 693	. . . . . . . 8 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → (◡𝐹 “ (V ∖ { 0 })) = ∅)
39	26, 29, 38	3eqtrd 2648	. . . . . . 7 ⊢ ((𝜑 ∧ ran 𝐹 ⊆ { 0 }) → 𝑊 = ∅)
40	39	ex 449	. . . . . 6 ⊢ (𝜑 → (ran 𝐹 ⊆ { 0 } → 𝑊 = ∅))
41	25, 40	sylbid 229	. . . . 5 ⊢ (𝜑 → (ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)} → 𝑊 = ∅))
42	41	necon3ad 2795	. . . 4 ⊢ (𝜑 → (𝑊 ≠ ∅ → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)}))
43	24, 42	mpd 15	. . 3 ⊢ (𝜑 → ¬ ran 𝐹 ⊆ {𝑧 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦)})
44	43	iffalsed 4047	. 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))))))
45		gsumval3a.i	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ ran ...)
46	45	iffalsed 4047	. 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)))))
47	23, 44, 46	3eqtrd 2648	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 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  ifcif 4036  {csn 4125  ^◡ccnv 5037  ran crn 5039   “ cima 5041   ∘ ccom 5042  ℩cio 5766   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   supp csupp 7182  Fincfn 7841  1c1 9816  ℤ_≥cuz 11563  ...cfz 12197  seqcseq 12663  #chash 12979  Basecbs 15695  +_gcplusg 15768  0_gc0g 15923   Σ_g cgsu 15924  Mndcmnd 17117  Cntzccntz 17571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118

This theorem is referenced by:  gsumval3lem2  18130