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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 12252 | . . . . 5 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀)) | |
6 | 4, 5 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘𝑀)) |
7 | peano2uz 12295 | . . . 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 12956 | . . . . . . 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 2827 | . . . . . . 7 ⊢ (𝜑 → (𝑀 + 1) = 𝑁) |
14 | 13 | preq2d 4670 | . . . . . 6 ⊢ (𝜑 → {𝑀, (𝑀 + 1)} = {𝑀, 𝑁}) |
15 | 11, 14 | eqtrd 2856 | . . . . 5 ⊢ (𝜑 → (𝑀...(𝑀 + 1)) = {𝑀, 𝑁}) |
16 | 15 | feq2d 6495 | . . . 4 ⊢ (𝜑 → (𝐹:(𝑀...(𝑀 + 1))⟶𝐵 ↔ 𝐹:{𝑀, 𝑁}⟶𝐵)) |
17 | 9, 16 | mpbird 259 | . . 3 ⊢ (𝜑 → 𝐹:(𝑀...(𝑀 + 1))⟶𝐵) |
18 | 1, 2, 3, 8, 17 | gsumval2 17890 | . 2 ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘(𝑀 + 1))) |
19 | seqp1 13378 | . . 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 13376 | . . . 4 ⊢ (𝑀 ∈ ℤ → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) | |
22 | 4, 21 | syl 17 | . . 3 ⊢ (𝜑 → (seq𝑀( + , 𝐹)‘𝑀) = (𝐹‘𝑀)) |
23 | 13 | fveq2d 6669 | . . 3 ⊢ (𝜑 → (𝐹‘(𝑀 + 1)) = (𝐹‘𝑁)) |
24 | 22, 23 | oveq12d 7168 | . 2 ⊢ (𝜑 → ((seq𝑀( + , 𝐹)‘𝑀) + (𝐹‘(𝑀 + 1))) = ((𝐹‘𝑀) + (𝐹‘𝑁))) |
25 | 18, 20, 24 | 3eqtrd 2860 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘𝑀) + (𝐹‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 {cpr 4563 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 1c1 10532 + caddc 10534 ℤcz 11975 ℤ≥cuz 12237 ...cfz 12886 seqcseq 13363 Basecbs 16477 +gcplusg 16559 Σg cgsu 16708 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 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-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-seq 13364 df-0g 16709 df-gsum 16710 |
This theorem is referenced by: gsumpr12val 17893 |
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