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Mirrors > Home > MPE Home > Th. List > gsumval | Structured version Visualization version GIF version |
Description: Expand out the substitutions in df-gsum 17427. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
gsumval.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumval.z | ⊢ 0 = (0g‘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) |
Ref | Expression |
---|---|
gsumval | ⊢ (𝜑 → (𝐺 Σg 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))))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumval.b | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumval.z | . 2 ⊢ 0 = (0g‘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 6910 | . . . 4 ⊢ 𝐵 ∈ V |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | fex2 7942 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | |
12 | 7, 8, 10, 11 | syl3anc 1368 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
13 | 7 | fdmd 6733 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
14 | 1, 2, 3, 4, 5, 6, 12, 13 | gsumvalx 18639 | 1 ⊢ (𝜑 → (𝐺 Σg 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 |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 {crab 3418 Vcvv 3461 ∖ cdif 3941 ⊆ wss 3944 ifcif 4530 ◡ccnv 5677 ran crn 5679 “ cima 5681 ∘ ccom 5682 ℩cio 6499 ⟶wf 6545 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 1c1 11141 ℤ≥cuz 12855 ...cfz 13519 seqcseq 14002 ♯chash 14325 Basecbs 17183 +gcplusg 17236 0gc0g 17424 Σg cgsu 17425 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-seq 14003 df-gsum 17427 |
This theorem is referenced by: gsumress 18645 gsumval1 18646 gsumval2a 18648 gsumval3a 19870 |
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