Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressmulgnn | Structured version Visualization version GIF version |
Description: Values for the group multiple function in a restricted structure. (Contributed by Thierry Arnoux, 12-Jun-2017.) |
Ref | Expression |
---|---|
ressmulgnn.1 | ⊢ 𝐻 = (𝐺 ↾s 𝐴) |
ressmulgnn.2 | ⊢ 𝐴 ⊆ (Base‘𝐺) |
ressmulgnn.3 | ⊢ ∗ = (.g‘𝐺) |
ressmulgnn.4 | ⊢ 𝐼 = (invg‘𝐺) |
Ref | Expression |
---|---|
ressmulgnn | ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressmulgnn.2 | . . . 4 ⊢ 𝐴 ⊆ (Base‘𝐺) | |
2 | ressmulgnn.1 | . . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
3 | eqid 2736 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 2, 3 | ressbas2 17046 | . . . 4 ⊢ (𝐴 ⊆ (Base‘𝐺) → 𝐴 = (Base‘𝐻)) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ 𝐴 = (Base‘𝐻) |
6 | eqid 2736 | . . 3 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
7 | eqid 2736 | . . 3 ⊢ (.g‘𝐻) = (.g‘𝐻) | |
8 | fvex 6838 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
9 | 8, 1 | ssexi 5266 | . . . . 5 ⊢ 𝐴 ∈ V |
10 | eqid 2736 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | 2, 10 | ressplusg 17097 | . . . . 5 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
12 | 9, 11 | ax-mp 5 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐻) |
13 | seqeq2 13826 | . . . 4 ⊢ ((+g‘𝐺) = (+g‘𝐻) → seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋}))) | |
14 | 12, 13 | ax-mp 5 | . . 3 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐻), (ℕ × {𝑋})) |
15 | 5, 6, 7, 14 | mulgnn 18804 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
16 | simpr 485 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
17 | 1, 16 | sselid 3930 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (Base‘𝐺)) |
18 | ressmulgnn.3 | . . . 4 ⊢ ∗ = (.g‘𝐺) | |
19 | eqid 2736 | . . . 4 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
20 | 3, 10, 18, 19 | mulgnn 18804 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (𝑁 ∗ 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
21 | 17, 20 | syldan 591 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁 ∗ 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
22 | 15, 21 | eqtr4d 2779 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ⊆ wss 3898 {csn 4573 × cxp 5618 ‘cfv 6479 (class class class)co 7337 1c1 10973 ℕcn 12074 seqcseq 13822 Basecbs 17009 ↾s cress 17038 +gcplusg 17059 invgcminusg 18674 .gcmg 18796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-n0 12335 df-z 12421 df-uz 12684 df-seq 13823 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulg 18797 |
This theorem is referenced by: ressmulgnn0 31580 |
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