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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 12276 | . . . . 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 2821 | . . . . 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 6768 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑁)) → (𝐹‘𝑖) = 𝑋) |
11 | elfznn 12933 | . . . . 5 ⊢ (𝑖 ∈ (1...𝑁) → 𝑖 ∈ ℕ) | |
12 | fvconst2g 6957 | . . . . 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 2858 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝑁)) → (𝐹‘𝑖) = ((ℕ × {𝑋})‘𝑖)) |
15 | 3, 14 | seqfveq 13391 | . 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 2820 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
18 | elfvex 6696 | . . . . 5 ⊢ (𝑋 ∈ (Base‘𝐺) → 𝐺 ∈ V) | |
19 | 18, 16 | eleq2s 2930 | . . . 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 6871 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → 𝐹:(1...𝑁)⟶𝐵) |
23 | 16, 17, 20, 3, 22 | gsumval2 17889 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg 𝐹) = (seq1((+g‘𝐺), 𝐹)‘𝑁)) |
24 | mulgnngsum.t | . . 3 ⊢ · = (.g‘𝐺) | |
25 | eqid 2820 | . . 3 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
26 | 16, 17, 24, 25 | mulgnn 18225 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
27 | 15, 23, 26 | 3eqtr4rd 2866 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 {csn 4560 ↦ cmpt 5139 × cxp 5546 ‘cfv 6348 (class class class)co 7149 1c1 10531 ℕcn 11631 ℤ≥cuz 12237 ...cfz 12889 seqcseq 13366 Basecbs 16476 +gcplusg 16558 Σg cgsu 16707 .gcmg 18217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-seq 13367 df-0g 16708 df-gsum 16709 df-mulg 18218 |
This theorem is referenced by: mulgnn0gsum 18227 |
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