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Mirrors > Home > MPE Home > Th. List > gsumprval | Structured version Visualization version GIF version |
Description: Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.) |
Ref | Expression |
---|---|
gsumprval.b | ⊢ 𝐵 = (Base‘𝐺) |
gsumprval.p | ⊢ + = (+g‘𝐺) |
gsumprval.g | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
gsumprval.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
gsumprval.n | ⊢ (𝜑 → 𝑁 = (𝑀 + 1)) |
gsumprval.f | ⊢ (𝜑 → 𝐹:{𝑀, 𝑁}⟶𝐵) |
Ref | Expression |
---|---|
gsumprval | ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘𝑀) + (𝐹‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gsumprval.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | gsumprval.p | . . 3 ⊢ + = (+g‘𝐺) | |
3 | gsumprval.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
4 | gsumprval.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | uzid 12246 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
7 | peano2uz 12289 | . . . 4 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → (𝑀 + 1) ∈ (ℤ≥‘𝑀)) |
9 | gsumprval.f | . . . 4 ⊢ (𝜑 → 𝐹:{𝑀, 𝑁}⟶𝐵) | |
10 | fzpr 12957 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) | |
11 | 4, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) |
12 | gsumprval.n | . . . . . . . 8 ⊢ (𝜑 → 𝑁 = (𝑀 + 1)) | |
13 | 12 | eqcomd 2804 | . . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) = 𝑁) |
14 | 13 | preq2d 4636 | . . . . . 6 ⊢ (𝜑 → {𝑀, (𝑀 + 1)} = {𝑀, 𝑁}) |
15 | 11, 14 | eqtrd 2833 | . . . . 5 ⊢ (𝜑 → (𝑀...(𝑀 + 1)) = {𝑀, 𝑁}) |
16 | 15 | feq2d 6473 | . . . 4 ⊢ (𝜑 → (𝐹:(𝑀...(𝑀 + 1))⟶𝐵 ↔ 𝐹:{𝑀, 𝑁}⟶𝐵)) |
17 | 9, 16 | mpbird 260 | . . 3 ⊢ (𝜑 → 𝐹:(𝑀...(𝑀 + 1))⟶𝐵) |
18 | 1, 2, 3, 8, 17 | gsumval2 17888 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘(𝑀 + 1))) |
19 | seqp1 13379 | . . 3 ⊢ (𝑀 ∈ (ℤ≥‘𝑀) → (seq𝑀( + , 𝐹)‘(𝑀 + 1)) = ((seq𝑀( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1)))) | |
20 | 6, 19 | syl 17 | . 2 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘(𝑀 + 1)) = ((seq𝑀( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1)))) |
21 | seq1 13377 | . . . 4 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) | |
22 | 4, 21 | syl 17 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
23 | 13 | fveq2d 6649 | . . 3 ⊢ (𝜑 → (𝐹‘(𝑀 + 1)) = (𝐹‘𝑁)) |
24 | 22, 23 | oveq12d 7153 | . 2 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))) = ((𝐹‘𝑀) + (𝐹‘𝑁))) |
25 | 18, 20, 24 | 3eqtrd 2837 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘𝑀) + (𝐹‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {cpr 4527 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 1c1 10527 + caddc 10529 ℤcz 11969 ℤ≥cuz 12231 ...cfz 12885 seqcseq 13364 Basecbs 16475 +gcplusg 16557 Σg cgsu 16706 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-0g 16707 df-gsum 16708 |
This theorem is referenced by: gsumpr12val 17891 |
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