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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 16417. (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 6426 | . . . 4 ⊢ 𝐵 ∈ V |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | fex2 7357 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | |
12 | 7, 8, 10, 11 | syl3anc 1491 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
13 | 7 | fdmd 6266 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
14 | 1, 2, 3, 4, 5, 6, 12, 13 | gsumvalx 17584 | 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 385 = wceq 1653 ∃wex 1875 ∈ wcel 2157 ∀wral 3090 ∃wrex 3091 {crab 3094 Vcvv 3386 ∖ cdif 3767 ⊆ wss 3770 ifcif 4278 ◡ccnv 5312 ran crn 5314 “ cima 5316 ∘ ccom 5317 ℩cio 6063 ⟶wf 6098 –1-1-onto→wf1o 6101 ‘cfv 6102 (class class class)co 6879 1c1 10226 ℤ≥cuz 11929 ...cfz 12579 seqcseq 13054 ♯chash 13369 Basecbs 16183 +gcplusg 16266 0gc0g 16414 Σg cgsu 16415 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-seq 13055 df-gsum 16417 |
This theorem is referenced by: gsumress 17590 gsumval1 17591 gsumval2a 17593 gsumval3a 18618 |
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