Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 487 | . . 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 30665 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
8 | 1, 2, 7 | syl2anc 586 | . 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 2821 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | 3, 10 | ressbas2 16549 | . . . . . . 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 2821 | . . . . . 6 ⊢ (.g‘𝐻) = (.g‘𝐻) | |
15 | 12, 13, 14 | mulg0 18225 | . . . . 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 487 | . . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → 𝑁 = 0) | |
18 | 17 | oveq1d 7165 | . . . 4 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (𝑁(.g‘𝐻)𝑋) = (0(.g‘𝐻)𝑋)) |
19 | 4, 9 | sseldi 3964 | . . . . 5 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → 𝑋 ∈ (Base‘𝐺)) |
20 | eqid 2821 | . . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
21 | 10, 20, 5 | mulg0 18225 | . . . . 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 2866 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (𝑁(.g‘𝐻)𝑋) = (0 ∗ 𝑋)) |
24 | 17 | oveq1d 7165 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (𝑁 ∗ 𝑋) = (0 ∗ 𝑋)) |
25 | 23, 24 | eqtr4d 2859 | . 2 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) ∧ 𝑁 = 0) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
26 | elnn0 11893 | . . . 4 ⊢ (𝑁 ∈ ℕ0 ↔ (𝑁 ∈ ℕ ∨ 𝑁 = 0)) | |
27 | 26 | biimpi 218 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
28 | 27 | adantr 483 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) → (𝑁 ∈ ℕ ∨ 𝑁 = 0)) |
29 | 8, 25, 28 | mpjaodan 955 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ‘cfv 6349 (class class class)co 7150 0cc0 10531 ℕcn 11632 ℕ0cn0 11891 Basecbs 16477 ↾s cress 16478 0gc0g 16707 invgcminusg 18098 .gcmg 18218 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulg 18219 |
This theorem is referenced by: xrge0mulgnn0 30671 |
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