Theorem gsumval1 18648

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 2728	. . 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 4078	. . . 4 ⊢ 𝑂 ⊆ 𝐵
10		fss 6742	. . . 4 ⊢ ((𝐹:𝐴⟶𝑂 ∧ 𝑂 ⊆ 𝐵) → 𝐹:𝐴⟶𝐵)
11	8, 9, 10	sylancl 584	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
12	1, 2, 3, 4, 5, 6, 7, 11	gsumval 18642	. 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 6732	. . 3 ⊢ (𝐹:𝐴⟶𝑂 → ran 𝐹 ⊆ 𝑂)
14		iftrue 4536	. . 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 2767	1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 )

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∀wral 3057  ∃wrex 3066  {crab 3428  Vcvv 3471   ∖ cdif 3944   ⊆ wss 3947  ifcif 4530  ^◡ccnv 5679  ran crn 5681   “ cima 5683   ∘ ccom 5684  ℩cio 6501  ⟶wf 6547  –1-1-onto→wf1o 6550  ‘cfv 6551  (class class class)co 7424  1c1 11145  ℤ_≥cuz 12858  ...cfz 13522  seqcseq 14004  ♯chash 14327  Basecbs 17185  +_gcplusg 17238  0_gc0g 17426   Σ_g cgsu 17427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-seq 14005  df-gsum 17429

This theorem is referenced by:  gsum0  18649  gsumval2  18651  gsumz  18793