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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 16715. (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 6683 | . . . 4 ⊢ 𝐵 ∈ V |
10 | 9 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
11 | fex2 7637 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ V) → 𝐹 ∈ V) | |
12 | 7, 8, 10, 11 | syl3anc 1367 | . 2 ⊢ (𝜑 → 𝐹 ∈ V) |
13 | 7 | fdmd 6522 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
14 | 1, 2, 3, 4, 5, 6, 12, 13 | gsumvalx 17885 | 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 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 ifcif 4466 ◡ccnv 5553 ran crn 5555 “ cima 5557 ∘ ccom 5558 ℩cio 6311 ⟶wf 6350 –1-1-onto→wf1o 6353 ‘cfv 6354 (class class class)co 7155 1c1 10537 ℤ≥cuz 12242 ...cfz 12891 seqcseq 13368 ♯chash 13689 Basecbs 16482 +gcplusg 16564 0gc0g 16712 Σg cgsu 16713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-seq 13369 df-gsum 16715 |
This theorem is referenced by: gsumress 17891 gsumval1 17892 gsumval2a 17894 gsumval3a 19022 |
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