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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 17331. (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 6861 | . . . 4 ⊢ 𝐵 ∈ V |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | fex2 7875 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | |
12 | 7, 8, 10, 11 | syl3anc 1372 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
13 | 7 | fdmd 6684 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
14 | 1, 2, 3, 4, 5, 6, 12, 13 | gsumvalx 18538 | 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 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∀wral 3065 ∃wrex 3074 {crab 3410 Vcvv 3448 ∖ cdif 3912 ⊆ wss 3915 ifcif 4491 ◡ccnv 5637 ran crn 5639 “ cima 5641 ∘ ccom 5642 ℩cio 6451 ⟶wf 6497 –1-1-onto→wf1o 6500 ‘cfv 6501 (class class class)co 7362 1c1 11059 ℤ≥cuz 12770 ...cfz 13431 seqcseq 13913 ♯chash 14237 Basecbs 17090 +gcplusg 17140 0gc0g 17328 Σg cgsu 17329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-seq 13914 df-gsum 17331 |
This theorem is referenced by: gsumress 18544 gsumval1 18545 gsumval2a 18547 gsumval3a 19687 |
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