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| Mirrors > Home > MPE Home > Th. List > mulgnn0p1 | Structured version Visualization version GIF version | ||
| Description: Group multiple (exponentiation) operation at a successor, extended to ℕ0. (Contributed by Mario Carneiro, 11-Dec-2014.) |
| Ref | Expression |
|---|---|
| mulgnn0p1.b | ⊢ 𝐵 = (Base‘𝐺) |
| mulgnn0p1.t | ⊢ · = (.g‘𝐺) |
| mulgnn0p1.p | ⊢ + = (+g‘𝐺) |
| Ref | Expression |
|---|---|
| mulgnn0p1 | ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpr 484 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
| 2 | simpl3 1194 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ 𝐵) | |
| 3 | mulgnn0p1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
| 4 | mulgnn0p1.t | . . . 4 ⊢ · = (.g‘𝐺) | |
| 5 | mulgnn0p1.p | . . . 4 ⊢ + = (+g‘𝐺) | |
| 6 | 3, 4, 5 | mulgnnp1 19100 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
| 7 | 1, 2, 6 | syl2anc 584 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
| 8 | eqid 2737 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 9 | 3, 5, 8 | mndlid 18767 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋) |
| 10 | 3, 8, 4 | mulg0 19092 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺)) |
| 11 | 10 | adantl 481 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = (0g‘𝐺)) |
| 12 | 11 | oveq1d 7446 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0 · 𝑋) + 𝑋) = ((0g‘𝐺) + 𝑋)) |
| 13 | 3, 4 | mulg1 19099 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
| 14 | 13 | adantl 481 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = 𝑋) |
| 15 | 9, 12, 14 | 3eqtr4rd 2788 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
| 16 | 15 | 3adant2 1132 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
| 17 | oveq1 7438 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
| 18 | 1e0p1 12775 | . . . . . . 7 ⊢ 1 = (0 + 1) | |
| 19 | 17, 18 | eqtr4di 2795 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1) |
| 20 | 19 | oveq1d 7446 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = (1 · 𝑋)) |
| 21 | oveq1 7438 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
| 22 | 21 | oveq1d 7446 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 · 𝑋) + 𝑋) = ((0 · 𝑋) + 𝑋)) |
| 23 | 20, 22 | eqeq12d 2753 | . . . 4 ⊢ (𝑁 = 0 → (((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋) ↔ (1 · 𝑋) = ((0 · 𝑋) + 𝑋))) |
| 24 | 16, 23 | syl5ibrcom 247 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))) |
| 25 | 24 | imp 406 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 = 0) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
| 26 | simp2 1138 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
| 27 | elnn0 12528 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
| 28 | 26, 27 | sylib 218 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
| 29 | 7, 25, 28 | mpjaodan 961 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 ℕcn 12266 ℕ0cn0 12526 Basecbs 17247 +gcplusg 17297 0gc0g 17484 Mndcmnd 18747 .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-seq 14043 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-mulg 19086 |
| This theorem is referenced by: mulgaddcom 19116 mulginvcom 19117 mulgneg2 19126 mhmmulg 19133 srgmulgass 20214 srgpcomp 20215 srgpcompp 20216 srgbinomlem4 20226 srgbinomlem 20227 lmodvsmmulgdi 20895 cnfldmulg 21416 cnfldexp 21417 assamulgscmlem2 21920 mplcoe3 22056 mhppwdeg 22154 psdpw 22174 tmdmulg 24100 clmmulg 25134 omndmul 33091 rprmdvdspow 33561 primrootsunit1 42098 aks6d1c1p6 42115 idomnnzpownz 42133 deg1pow 42142 unitscyglem5 42200 domnexpgn0cl 42533 abvexp 42542 lmodvsmdi 48295 |
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