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Mirrors > Home > ILE Home > Th. List > mulgnn0p1 | 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 110 | . . 3 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
2 | simpl3 1002 | . . 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 12867 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
7 | 1, 2, 6 | syl2anc 411 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
8 | eqid 2177 | . . . . . . 7 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
9 | 3, 5, 8 | mndlid 12715 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0g‘𝐺) + 𝑋) = 𝑋) |
10 | 3, 8, 4 | mulg0 12864 | . . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = (0g‘𝐺)) |
11 | 10 | adantl 277 | . . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (0 · 𝑋) = (0g‘𝐺)) |
12 | 11 | oveq1d 5883 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → ((0 · 𝑋) + 𝑋) = ((0g‘𝐺) + 𝑋)) |
13 | 3, 4 | mulg1 12866 | . . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋) |
14 | 13 | adantl 277 | . . . . . 6 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = 𝑋) |
15 | 9, 12, 14 | 3eqtr4rd 2221 | . . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
16 | 15 | 3adant2 1016 | . . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (1 · 𝑋) = ((0 · 𝑋) + 𝑋)) |
17 | oveq1 5875 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝑁 + 1) = (0 + 1)) | |
18 | 1e0p1 9401 | . . . . . . 7 ⊢ 1 = (0 + 1) | |
19 | 17, 18 | eqtr4di 2228 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 + 1) = 1) |
20 | 19 | oveq1d 5883 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = (1 · 𝑋)) |
21 | oveq1 5875 | . . . . . 6 ⊢ (𝑁 = 0 → (𝑁 · 𝑋) = (0 · 𝑋)) | |
22 | 21 | oveq1d 5883 | . . . . 5 ⊢ (𝑁 = 0 → ((𝑁 · 𝑋) + 𝑋) = ((0 · 𝑋) + 𝑋)) |
23 | 20, 22 | eqeq12d 2192 | . . . 4 ⊢ (𝑁 = 0 → (((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋) ↔ (1 · 𝑋) = ((0 · 𝑋) + 𝑋))) |
24 | 16, 23 | syl5ibrcom 157 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 = 0 → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋))) |
25 | 24 | imp 124 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 = 0) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
26 | simp2 998 | . . 3 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → 𝑁 ∈ ℕ0) | |
27 | elnn0 9154 | . . 3 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
28 | 26, 27 | sylib 122 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
29 | 7, 25, 28 | mpjaodan 798 | 1 ⊢ ((𝐺 ∈ Mnd ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑁 + 1) · 𝑋) = ((𝑁 · 𝑋) + 𝑋)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5211 (class class class)co 5868 0cc0 7789 1c1 7790 + caddc 7792 ℕcn 8895 ℕ0cn0 9152 Basecbs 12432 +gcplusg 12505 0gc0g 12640 Mndcmnd 12696 .gcmg 12859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-addcom 7889 ax-addass 7891 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-2 8954 df-n0 9153 df-z 9230 df-uz 9505 df-seqfrec 10419 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-minusg 12758 df-mulg 12860 |
This theorem is referenced by: mulgaddcom 12882 mulginvcom 12883 mulgneg2 12892 mhmmulg 12899 srgmulgass 12985 srgpcomp 12986 srgpcompp 12987 |
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