Theorem gsumval1 18591

Description: Value of the group sum operation when every element being summed is an identity of 𝐺. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
gsumval1.b	⊢ 𝐵 = (Base‘𝐺)
gsumval1.z	⊢ 0 = (0g‘𝐺)
gsumval1.p	⊢ + = (+g‘𝐺)
gsumval1.o	⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
gsumval1.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
gsumval1.a	⊢ (𝜑 → 𝐴 ∈ 𝑊)
gsumval1.f	⊢ (𝜑 → 𝐹:𝐴⟶𝑂)

Assertion

Ref	Expression
gsumval1	⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 )

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥, + ,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑂(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem gsumval1

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

Step	Hyp	Ref	Expression
1		gsumval1.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		gsumval1.z	. . 3 ⊢ 0 = (0g‘𝐺)
3		gsumval1.p	. . 3 ⊢ + = (+g‘𝐺)
4		gsumval1.o	. . 3 ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5		eqidd 2732	. . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ 𝑂)) = (◡𝐹 “ (V ∖ 𝑂)))
6		gsumval1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
7		gsumval1.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
8		gsumval1.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝑂)
9	4	ssrab3 4029	. . . 4 ⊢ 𝑂 ⊆ 𝐵
10		fss 6667	. . . 4 ⊢ ((𝐹:𝐴⟶𝑂 ∧ 𝑂 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
11	8, 9, 10	sylancl 586	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12	1, 2, 3, 4, 5, 6, 7, 11	gsumval 18585	. 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂)))))))))
13		frn 6658	. . 3 ⊢ (𝐹:𝐴⟶𝑂 → ran 𝐹 ⊆ 𝑂)
14		iftrue 4478	. . 3 ⊢ (ran 𝐹 ⊆ 𝑂 → if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂)))))))) = 0 )
15	8, 13, 14	3syl 18	. 2 ⊢ (𝜑 → if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(♯‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘(◡𝐹 “ (V ∖ 𝑂)))))))) = 0 )
16	12, 15	eqtrd 2766	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ∖ cdif 3894   ⊆ wss 3897  ifcif 4472  ^◡ccnv 5613  ran crn 5615   “ cima 5617   ∘ ccom 5618  ℩cio 6435  ⟶wf 6477  –1-1-onto→wf1o 6480  ‘cfv 6481  (class class class)co 7346  1c1 11007  ℤ_≥cuz 12732  ...cfz 13407  seqcseq 13908  ♯chash 14237  Basecbs 17120  +_gcplusg 17161  0_gc0g 17343   Σ_g cgsu 17344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-seq 13909  df-gsum 17346

This theorem is referenced by:  gsum0  18592  gsumval2  18594  gsumz  18744