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 2610 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
4 | 2, 3 | ressbas2 15758 | . . . 4 ⊢ (𝐴 ⊆ (Base‘𝐺) → 𝐴 = (Base‘𝐻)) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ 𝐴 = (Base‘𝐻) |
6 | eqid 2610 | . . 3 ⊢ (+g‘𝐻) = (+g‘𝐻) | |
7 | eqid 2610 | . . 3 ⊢ (.g‘𝐻) = (.g‘𝐻) | |
8 | fvex 6113 | . . . . . 6 ⊢ (Base‘𝐺) ∈ V | |
9 | 8, 1 | ssexi 4731 | . . . . 5 ⊢ 𝐴 ∈ V |
10 | eqid 2610 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | 2, 10 | ressplusg 15818 | . . . . 5 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
12 | 9, 11 | ax-mp 5 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐻) |
13 | seqeq2 12667 | . . . 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 17370 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
16 | simpr 476 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴) | |
17 | 1, 16 | sseldi 3566 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ (Base‘𝐺)) |
18 | ressmulgnn.3 | . . . 4 ⊢ ∗ = (.g‘𝐺) | |
19 | eqid 2610 | . . . 4 ⊢ seq1((+g‘𝐺), (ℕ × {𝑋})) = seq1((+g‘𝐺), (ℕ × {𝑋})) | |
20 | 3, 10, 18, 19 | mulgnn 17370 | . . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (Base‘𝐺)) → (𝑁 ∗ 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
21 | 17, 20 | syldan 486 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁 ∗ 𝑋) = (seq1((+g‘𝐺), (ℕ × {𝑋}))‘𝑁)) |
22 | 15, 21 | eqtr4d 2647 | 1 ⊢ ((𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐴) → (𝑁(.g‘𝐻)𝑋) = (𝑁 ∗ 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {csn 4125 × cxp 5036 ‘cfv 5804 (class class class)co 6549 1c1 9816 ℕcn 10897 seqcseq 12663 Basecbs 15695 ↾s cress 15696 +gcplusg 15768 invgcminusg 17246 .gcmg 17363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-z 11255 df-seq 12664 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulg 17364 |
This theorem is referenced by: ressmulgnn0 29015 |
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