Theorem gsumval1 18743

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 2770	. . 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 4044	. . . 4 ⊢ 𝑂 ⊆ 𝐵
10		fss 6725	. . . 4 ⊢ ((𝐹:𝐴⟶𝑂 ∧ 𝑂 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
11	8, 9, 10	sylancl 597	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12	1, 2, 3, 4, 5, 6, 7, 11	gsumval 18737	. 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 6716	. . 3 ⊢ (𝐹:𝐴⟶𝑂 → ran 𝐹 ⊆ 𝑂)
14		iftrue 4498	. . 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 19	. 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 2804	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 )

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ∖ cdif 3910   ⊆ wss 3913  ifcif 4492  ^◡ccnv 5663  ran crn 5665   “ cima 5667   ∘ ccom 5668  ℩cio 6493  ⟶wf 6535  –1-1-onto→wf1o 6538  ‘cfv 6539  (class class class)co 7413  1c1 11103  ℤ_≥cuz 12864  ...cfz 13537  seqcseq 14039  ♯chash 14368  Basecbs 17271  +_gcplusg 17312  0_gc0g 17494   Σ_g cgsu 17495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7416  df-oprab 7417  df-mpo 7418  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-seq 14040  df-gsum 17497

This theorem is referenced by:  gsum0  18744  gsumval2  18746  gsumz  18897