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Mirrors > Home > MPE Home > Th. List > ressmulgnn0 | Structured version Visualization version GIF version |
Description: Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 14-Jun-2017.) |
Ref | Expression |
---|---|
ressmulgnn.1 | ⊢ 𝐻 = (𝐺 ↾s 𝐴) |
ressmulgnn.2 | ⊢ 𝐴 ⊆ (Base‘𝐺) |
ressmulgnn.3 | ⊢ ∗ = (.g‘𝐺) |
ressmulgnn.4 | ⊢ 𝐼 = (invg‘𝐺) |
ressmulgnn0.4 | ⊢ (0g‘𝐺) = (0g‘𝐻) |
Ref | Expression |
---|---|
ressmulgnn0 | ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 483 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
2 | simplr 767 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 ∈ ℕ) → 𝑋 ∈ 𝐴) | |
3 | ressmulgnn.1 | . . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
4 | ressmulgnn.2 | . . . 4 ⊢ 𝐴 ⊆ (Base‘𝐺) | |
5 | ressmulgnn.3 | . . . 4 ⊢ ∗ = (.g‘𝐺) | |
6 | ressmulgnn.4 | . . . 4 ⊢ 𝐼 = (invg‘𝐺) | |
7 | 3, 4, 5, 6 | ressmulgnn 19070 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
8 | 1, 2, 7 | syl2anc 582 | . 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 ∈ ℕ) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
9 | simplr 767 | . . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → 𝑋 ∈ 𝐴) | |
10 | eqid 2726 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | 3, 10 | ressbas2 17251 | . . . . . . 7 ⊢ (𝐴 ⊆ (Base‘𝐺) → 𝐴 = (Base‘𝐻)) |
12 | 4, 11 | ax-mp 5 | . . . . . 6 ⊢ 𝐴 = (Base‘𝐻) |
13 | ressmulgnn0.4 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐻) | |
14 | eqid 2726 | . . . . . 6 ⊢ (.g‘𝐻) = (.g‘𝐻) | |
15 | 12, 13, 14 | mulg0 19068 | . . . . 5 ⊢ (𝑋 ∈ 𝐴 → (0(.g‘𝐻)𝑋) = (0g‘𝐺)) |
16 | 9, 15 | syl 17 | . . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (0(.g‘𝐻)𝑋) = (0g‘𝐺)) |
17 | simpr 483 | . . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → 𝑁 = 0) | |
18 | 17 | oveq1d 7439 | . . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (𝑁(.g‘𝐻)𝑋) = (0(.g‘𝐻)𝑋)) |
19 | 4, 9 | sselid 3977 | . . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → 𝑋 ∈ (Base‘𝐺)) |
20 | eqid 2726 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
21 | 10, 20, 5 | mulg0 19068 | . . . . 5 ⊢ (𝑋 ∈ (Base‘𝐺) → (0 ∗ 𝑋) = (0g‘𝐺)) |
22 | 19, 21 | syl 17 | . . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (0 ∗ 𝑋) = (0g‘𝐺)) |
23 | 16, 18, 22 | 3eqtr4d 2776 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (𝑁(.g‘𝐻)𝑋) = (0 ∗ 𝑋)) |
24 | 17 | oveq1d 7439 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (𝑁 ∗ 𝑋) = (0 ∗ 𝑋)) |
25 | 23, 24 | eqtr4d 2769 | . 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
26 | elnn0 12526 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
27 | 26 | biimpi 215 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
28 | 27 | adantr 479 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
29 | 8, 25, 28 | mpjaodan 956 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ‘cfv 6554 (class class class)co 7424 0cc0 11158 ℕcn 12264 ℕ0cn0 12524 Basecbs 17213 ↾s cress 17242 0gc0g 17454 invgcminusg 18929 .gcmg 19061 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-n0 12525 df-z 12611 df-uz 12875 df-seq 14022 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulg 19062 |
This theorem is referenced by: fermltlchr 21523 xrge0mulgnn0 32898 znfermltl 33241 |
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