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Theorem gsumval 18872

Description: Expand out the substitutions in df-gsum 17619. (Contributed by Mario Carneiro, 7-Dec-2014.)

**Hypotheses**
Ref	Expression
gsumval.b	⊢ 𝐵 = (Base‘𝐺)
gsumval.z	⊢ 0 = (0_g‘𝐺)
gsumval.p	⊢ + = (+_g‘𝐺)
gsumval.o	⊢ 𝑂 = {𝑠 ∈ 𝐵 ∣ ∀𝑡 ∈ 𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)}
gsumval.w	⊢ (𝜑 → 𝑊 = (^◡𝐹 “ (V ∖ 𝑂)))
gsumval.g	⊢ (𝜑 → 𝐺 ∈ 𝑉)
gsumval.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
gsumval.f	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

**Assertion**
Ref	Expression
gsumval	⊢ (𝜑 → (𝐺 Σ_g 𝐹) = if(ran 𝐹 ⊆ 𝑂, 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)))))))

Distinct variable groups: 𝑡,𝑠,𝑥,𝐵 𝑓,𝑚,𝑛,𝑥,𝜑 𝑓,𝐹,𝑚,𝑛,𝑥 𝑓,𝐺,𝑚,𝑛,𝑥 + ,𝑠,𝑡,𝑥 𝑓,𝑂,𝑚,𝑛,𝑥 Allowed substitution hints: 𝜑(𝑡, 𝑠) 𝐴(𝑥, 𝑡, 𝑓, 𝑚, 𝑛, 𝑠) 𝐵(𝑓, 𝑚, 𝑛) + (𝑓, 𝑚, 𝑛) 𝐹(𝑡, 𝑠) 𝐺(𝑡, 𝑠) 𝑂(𝑡, 𝑠) 𝑉(𝑥, 𝑡, 𝑓, 𝑚, 𝑛, 𝑠) 𝑊(𝑥, 𝑡, 𝑓, 𝑚, 𝑛, 𝑠) 𝑋(𝑥, 𝑡, 𝑓, 𝑚, 𝑛, 𝑠) 0 (𝑥, 𝑡, 𝑓, 𝑚, 𝑛, 𝑠)

**Proof of Theorem gsumval**
Step	Hyp	Ref	Expression
1		gsumval.b	. 2 ⊢ 𝐵 = (Base‘𝐺)
2		gsumval.z	. 2 ⊢ 0 = (0_g‘𝐺)
3		gsumval.p	. 2 ⊢ + = (+_g‘𝐺)
4		gsumval.o	. 2 ⊢ 𝑂 = {𝑠 ∈ 𝐵 ∣ ∀𝑡 ∈ 𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)}
5		gsumval.w	. 2 ⊢ (𝜑 → 𝑊 = (^◡𝐹 “ (V ∖ 𝑂)))
6		gsumval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝑉)
7		gsumval.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
8		gsumval.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
9	1	fvexi 6900	. . . 4 ⊢ 𝐵 ∈ V
10	9	a1i 11	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
11		fex2 7949	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ V) → 𝐹 ∈ V)
12	7, 8, 10, 11	syl3anc 1398	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
13	7	fdmd 6721	. 2 ⊢ (𝜑 → dom 𝐹 = 𝐴)
14	1, 2, 3, 4, 5, 6, 12, 13	gsumvalx 18871	1 ⊢ (𝜑 → (𝐺 Σ_g 𝐹) = if(ran 𝐹 ⊆ 𝑂, 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)))))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∃wex 1811 ∈ wcel 2144 ∀wral 3076 ∃wrex 3086 {crab 3412 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ifcif 4481 ^◡ccnv 5649 ran crn 5651 “ cima 5653 ∘ ccom 5654 ℩cio 6492 ⟶wf 6535 –1-1-onto→wf1o 6538 ‘cfv 6539 (class class class)co 7421 1c1 11205 ℤ_≥cuz 12969 ...cfz 13643 seqcseq 14148 ♯chash 14478 Basecbs 17393 +_gcplusg 17434 0_gc0g 17616 Σ_g cgsu 17617

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7424 df-oprab 7425 df-mpo 7426 df-frecs 8303 df-wrecs 8334 df-recs 8383 df-rdg 8422 df-seq 14149 df-gsum 17619

This theorem is used by: gsumress 18877 gsumval1 18878 gsumval2a 18880 gsumval3a 20123

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