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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 11978 | . . 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 18351 | . . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
6 | 5 | ex 416 | . . . 4 ⊢ (𝑁 ∈ ℕ → (𝑋 ∈ 𝐵 → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))) |
7 | oveq1 7177 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
8 | eqid 2738 | . . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | 2, 8, 3 | mulg0 18349 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺)) |
10 | 7, 9 | sylan9eq 2793 | . . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (0g‘𝐺)) |
11 | oveq2 7178 | . . . . . . . . . . . . 13 ⊢ (𝑁 = 0 → (1...𝑁) = (1...0)) | |
12 | fz10 13019 | . . . . . . . . . . . . 13 ⊢ (1...0) = ∅ | |
13 | 11, 12 | eqtrdi 2789 | . . . . . . . . . . . 12 ⊢ (𝑁 = 0 → (1...𝑁) = ∅) |
14 | eqidd 2739 | . . . . . . . . . . . 12 ⊢ (𝑁 = 0 → 𝑋 = 𝑋) | |
15 | 13, 14 | mpteq12dv 5115 | . . . . . . . . . . 11 ⊢ (𝑁 = 0 → (𝑥 ∈ (1...𝑁) ↦ 𝑋) = (𝑥 ∈ ∅ ↦ 𝑋)) |
16 | mpt0 6479 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ ∅ ↦ 𝑋) = ∅ | |
17 | 15, 16 | eqtrdi 2789 | . . . . . . . . . 10 ⊢ (𝑁 = 0 → (𝑥 ∈ (1...𝑁) ↦ 𝑋) = ∅) |
18 | 4, 17 | syl5eq 2785 | . . . . . . . . 9 ⊢ (𝑁 = 0 → 𝐹 = ∅) |
19 | 18 | adantr 484 | . . . . . . . 8 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → 𝐹 = ∅) |
20 | 19 | oveq2d 7186 | . . . . . . 7 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg 𝐹) = (𝐺 Σg ∅)) |
21 | 8 | gsum0 18010 | . . . . . . 7 ⊢ (𝐺 Σg ∅) = (0g‘𝐺) |
22 | 20, 21 | eqtrdi 2789 | . . . . . 6 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → (𝐺 Σg 𝐹) = (0g‘𝐺)) |
23 | 10, 22 | eqtr4d 2776 | . . . . 5 ⊢ ((𝑁 = 0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
24 | 23 | ex 416 | . . . 4 ⊢ (𝑁 = 0 → (𝑋 ∈ 𝐵 → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))) |
25 | 6, 24 | jaoi 856 | . . 3 ⊢ ((𝑁 ∈ ℕ ∨ 𝑁 = 0) → (𝑋 ∈ 𝐵 → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))) |
26 | 1, 25 | sylbi 220 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑋 ∈ 𝐵 → (𝑁 · 𝑋) = (𝐺 Σg 𝐹))) |
27 | 26 | imp 410 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 · 𝑋) = (𝐺 Σg 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2114 ∅c0 4211 ↦ cmpt 5110 ‘cfv 6339 (class class class)co 7170 0cc0 10615 1c1 10616 ℕcn 11716 ℕ0cn0 11976 ...cfz 12981 Basecbs 16586 0gc0g 16816 Σg cgsu 16817 .gcmg 18342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-seq 13461 df-0g 16818 df-gsum 16819 df-mulg 18343 |
This theorem is referenced by: (None) |
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