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| Mirrors > Home > MPE Home > Th. List > mulgnn0gsum | Structured version Visualization version GIF version | ||
| Description: Group multiple (exponentiation) operation at a nonnegative integer expressed by a group sum. This corresponds to the definition in [Lang] p. 6, second formula. (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 |
|---|---|
| mulgnn0gsum | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elnn0 12528 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
| 2 | mulgnngsum.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | mulgnngsum.t | . . . . . 6 ⊢ · = (.g‘𝐺) | |
| 4 | mulgnngsum.f | . . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (1...𝑁) ↦ 𝑋) | |
| 5 | 2, 3, 4 | mulgnngsum 19097 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
| 6 | 5 | ex 412 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑋 ∈ 𝐵 → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))) |
| 7 | oveq1 7438 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
| 8 | eqid 2737 | . . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 9 | 2, 8, 3 | mulg0 19092 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺)) |
| 10 | 7, 9 | sylan9eq 2797 | . . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (0g‘𝐺)) |
| 11 | oveq2 7439 | . . . . . . . . . . . . 13 ⊢ (𝑁 = 0 → (1...𝑁) = (1...0)) | |
| 12 | fz10 13585 | . . . . . . . . . . . . 13 ⊢ (1...0) = ∅ | |
| 13 | 11, 12 | eqtrdi 2793 | . . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (1...𝑁) = ∅) |
| 14 | eqidd 2738 | . . . . . . . . . . . 12 ⊢ (𝑁 = 0 → 𝑋 = 𝑋) | |
| 15 | 13, 14 | mpteq12dv 5233 | . . . . . . . . . . 11 ⊢ (𝑁 = 0 → (𝑥 ∈ (1...𝑁) ↦ 𝑋) = (𝑥 ∈ ∅ ↦ 𝑋)) |
| 16 | mpt0 6710 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ∅ ↦ 𝑋) = ∅ | |
| 17 | 15, 16 | eqtrdi 2793 | . . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑥 ∈ (1...𝑁) ↦ 𝑋) = ∅) |
| 18 | 4, 17 | eqtrid 2789 | . . . . . . . . 9 ⊢ (𝑁 = 0 → 𝐹 = ∅) |
| 19 | 18 | adantr 480 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → 𝐹 = ∅) |
| 20 | 19 | oveq2d 7447 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg 𝐹) = (𝐺 Σg ∅)) |
| 21 | 8 | gsum0 18697 | . . . . . . 7 ⊢ (𝐺 Σg ∅) = (0g‘𝐺) |
| 22 | 20, 21 | eqtrdi 2793 | . . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg 𝐹) = (0g‘𝐺)) |
| 23 | 10, 22 | eqtr4d 2780 | . . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
| 24 | 23 | ex 412 | . . . 4 ⊢ (𝑁 = 0 → (𝑋 ∈ 𝐵 → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))) |
| 25 | 6, 24 | jaoi 858 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑋 ∈ 𝐵 → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))) |
| 26 | 1, 25 | sylbi 217 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ 𝐵 → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))) |
| 27 | 26 | imp 406 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∅c0 4333 ↦ cmpt 5225 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ℕcn 12266 ℕ0cn0 12526 ...cfz 13547 Basecbs 17247 0gc0g 17484 Σg cgsu 17485 .gcmg 19085 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-0g 17486 df-gsum 17487 df-mulg 19086 |
| This theorem is referenced by: (None) |
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