Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mulgnngsum | Structured version Visualization version GIF version |
Description: Group multiple (exponentiation) operation at a positive integer expressed by a group sum. (Contributed by AV, 28-Dec-2023.) |
Ref | Expression |
---|---|
mulgnngsum.b | ⊢ 𝐵 = (Base‘𝐺) |
mulgnngsum.t | ⊢ · = (.g‘𝐺) |
mulgnngsum.f | ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋) |
Ref | Expression |
---|---|
mulgnngsum | ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnnuz 12283 | . . . . 5 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (ℤ≥‘1)) | |
2 | 1 | biimpi 218 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ (ℤ≥‘1)) |
3 | 2 | adantr 483 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ (ℤ≥‘1)) |
4 | mulgnngsum.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋) | |
5 | 4 | a1i 11 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑁)) → 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋)) |
6 | eqidd 2822 | . . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑁)) ∧ 𝑥 = 𝑖) → 𝑋 = 𝑋) | |
7 | simpr 487 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑁)) → 𝑖 ∈ (1...𝑁)) | |
8 | simpr 487 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
9 | 8 | adantr 483 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑁)) → 𝑋 ∈ 𝐵) |
10 | 5, 6, 7, 9 | fvmptd 6775 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑁)) → (𝐹‘𝑖) = 𝑋) |
11 | elfznn 12937 | . . . . 5 ⊢ (𝑖 ∈ (1...𝑁) → 𝑖 ∈ ℕ) | |
12 | fvconst2g 6964 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑖 ∈ ℕ) → ((ℕ × {𝑋})‘𝑖) = 𝑋) | |
13 | 8, 11, 12 | syl2an 597 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑁)) → ((ℕ × {𝑋})‘𝑖) = 𝑋) |
14 | 10, 13 | eqtr4d 2859 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑁)) → (𝐹‘𝑖) = ((ℕ × {𝑋})‘𝑖)) |
15 | 3, 14 | seqfveq 13395 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (seq1((+g‘𝐺), 𝐹)‘𝑁) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
16 | mulgnngsum.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
17 | eqid 2821 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
18 | elfvex 6703 | . . . . 5 ⊢ (𝑋 ∈ (Base‘𝐺) → 𝐺 ∈ V) | |
19 | 18, 16 | eleq2s 2931 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → 𝐺 ∈ V) |
20 | 19 | adantl 484 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → 𝐺 ∈ V) |
21 | 8 | adantr 483 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑥 ∈ (1...𝑁)) → 𝑋 ∈ 𝐵) |
22 | 21, 4 | fmptd 6878 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → 𝐹:(1...𝑁)⟶𝐵) |
23 | 16, 17, 20, 3, 22 | gsumval2 17896 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg 𝐹) = (seq1((+g‘𝐺), 𝐹)‘𝑁)) |
24 | mulgnngsum.t | . . 3 ⊢ · = (.g‘𝐺) | |
25 | eqid 2821 | . . 3 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
26 | 16, 17, 24, 25 | mulgnn 18232 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
27 | 15, 23, 26 | 3eqtr4rd 2867 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 ↦ cmpt 5146 × cxp 5553 ‘cfv 6355 (class class class)co 7156 1c1 10538 ℕcn 11638 ℤ≥cuz 12244 ...cfz 12893 seqcseq 13370 Basecbs 16483 +gcplusg 16565 Σg cgsu 16714 .gcmg 18224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-seq 13371 df-0g 16715 df-gsum 16716 df-mulg 18225 |
This theorem is referenced by: mulgnn0gsum 18234 |
Copyright terms: Public domain | W3C validator |