Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ressplusf | Structured version Visualization version GIF version |
Description: The group operation function +𝑓 of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.) |
Ref | Expression |
---|---|
ressplusf.1 | ⊢ 𝐵 = (Base‘𝐺) |
ressplusf.2 | ⊢ 𝐻 = (𝐺 ↾s 𝐴) |
ressplusf.3 | ⊢ ⨣ = (+g‘𝐺) |
ressplusf.4 | ⊢ ⨣ Fn (𝐵 × 𝐵) |
ressplusf.5 | ⊢ 𝐴 ⊆ 𝐵 |
Ref | Expression |
---|---|
ressplusf | ⊢ (+𝑓‘𝐻) = ( ⨣ ↾ (𝐴 × 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressplusf.5 | . . 3 ⊢ 𝐴 ⊆ 𝐵 | |
2 | resmpt2 6758 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦))) | |
3 | 1, 1, 2 | mp2an 708 | . 2 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦)) |
4 | ressplusf.4 | . . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵) | |
5 | fnov 6768 | . . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) ↔ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦))) | |
6 | 4, 5 | mpbi 220 | . . 3 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) |
7 | 6 | reseq1i 5392 | . 2 ⊢ ( ⨣ ↾ (𝐴 × 𝐴)) = ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) ↾ (𝐴 × 𝐴)) |
8 | ressplusf.2 | . . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
9 | ressplusf.1 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
10 | 8, 9 | ressbas2 15931 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝐻)) |
11 | 1, 10 | ax-mp 5 | . . 3 ⊢ 𝐴 = (Base‘𝐻) |
12 | ressplusf.3 | . . . 4 ⊢ ⨣ = (+g‘𝐺) | |
13 | fvex 6201 | . . . . . . 7 ⊢ (Base‘𝐺) ∈ V | |
14 | 9, 13 | eqeltri 2697 | . . . . . 6 ⊢ 𝐵 ∈ V |
15 | 14, 1 | ssexi 4803 | . . . . 5 ⊢ 𝐴 ∈ V |
16 | eqid 2622 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
17 | 8, 16 | ressplusg 15993 | . . . . 5 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
18 | 15, 17 | ax-mp 5 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐻) |
19 | 12, 18 | eqtri 2644 | . . 3 ⊢ ⨣ = (+g‘𝐻) |
20 | eqid 2622 | . . 3 ⊢ (+𝑓‘𝐻) = (+𝑓‘𝐻) | |
21 | 11, 19, 20 | plusffval 17247 | . 2 ⊢ (+𝑓‘𝐻) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦)) |
22 | 3, 7, 21 | 3eqtr4ri 2655 | 1 ⊢ (+𝑓‘𝐻) = ( ⨣ ↾ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 × cxp 5112 ↾ cres 5116 Fn wfn 5883 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 Basecbs 15857 ↾s cress 15858 +gcplusg 15941 +𝑓cplusf 17239 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-plusf 17241 |
This theorem is referenced by: xrge0pluscn 29986 xrge0tmdOLD 29991 |
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