Theorem gsumval1 18609

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 4081	. . . 4 ⊢ 𝑂 ⊆ 𝐵
10		fss 6735	. . . 4 ⊢ ((𝐹:𝐴⟶𝑂 ∧ 𝑂 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
11	8, 9, 10	sylancl 585	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12	1, 2, 3, 4, 5, 6, 7, 11	gsumval 18603	. 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 6725	. . 3 ⊢ (𝐹:𝐴⟶𝑂 → ran 𝐹 ⊆ 𝑂)
14		iftrue 4535	. . 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 2771	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1780   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  {crab 3431  Vcvv 3473   ∖ cdif 3946   ⊆ wss 3949  ifcif 4529  ^◡ccnv 5676  ran crn 5678   “ cima 5680   ∘ ccom 5681  ℩cio 6494  ⟶wf 6540  –1-1-onto→wf1o 6543  ‘cfv 6544  (class class class)co 7412  1c1 11114  ℤ_≥cuz 12827  ...cfz 13489  seqcseq 13971  ♯chash 14295  Basecbs 17149  +_gcplusg 17202  0_gc0g 17390   Σ_g cgsu 17391

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-seq 13972  df-gsum 17393

This theorem is referenced by:  gsum0  18610  gsumval2  18612  gsumz  18754