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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 12889 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
7 | peano2uz 12937 | . . . 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 13610 | . . . . . . 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 2732 | . . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) = 𝑁) |
14 | 13 | preq2d 4749 | . . . . . 6 ⊢ (𝜑 → {𝑀, (𝑀 + 1)} = {𝑀, 𝑁}) |
15 | 11, 14 | eqtrd 2766 | . . . . 5 ⊢ (𝜑 → (𝑀...(𝑀 + 1)) = {𝑀, 𝑁}) |
16 | 15 | feq2d 6714 | . . . 4 ⊢ (𝜑 → (𝐹:(𝑀...(𝑀 + 1))⟶𝐵 ↔ 𝐹:{𝑀, 𝑁}⟶𝐵)) |
17 | 9, 16 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝐹:(𝑀...(𝑀 + 1))⟶𝐵) |
18 | 1, 2, 3, 8, 17 | gsumval2 18679 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘(𝑀 + 1))) |
19 | seqp1 14036 | . . 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 14034 | . . . 4 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) | |
22 | 4, 21 | syl 17 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
23 | 13 | fveq2d 6905 | . . 3 ⊢ (𝜑 → (𝐹‘(𝑀 + 1)) = (𝐹‘𝑁)) |
24 | 22, 23 | oveq12d 7442 | . 2 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))) = ((𝐹‘𝑀) + (𝐹‘𝑁))) |
25 | 18, 20, 24 | 3eqtrd 2770 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘𝑀) + (𝐹‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 {cpr 4635 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 1c1 11159 + caddc 11161 ℤcz 12610 ℤ≥cuz 12874 ...cfz 13538 seqcseq 14021 Basecbs 17213 +gcplusg 17266 Σg cgsu 17455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-seq 14022 df-0g 17456 df-gsum 17457 |
This theorem is referenced by: gsumpr12val 18682 |
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