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 | resmpo 7388 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦))) | |
3 | 1, 1, 2 | mp2an 689 | . 2 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦)) |
4 | ressplusf.4 | . . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵) | |
5 | fnov 7399 | . . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) ↔ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦))) | |
6 | 4, 5 | mpbi 229 | . . 3 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) |
7 | 6 | reseq1i 5886 | . 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 16947 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘𝐻)) |
11 | 1, 10 | ax-mp 5 | . . 3 ⊢ 𝐴 = (Base‘𝐻) |
12 | ressplusf.3 | . . . 4 ⊢ ⨣ = (+g‘𝐺) | |
13 | 9 | fvexi 6785 | . . . . . 6 ⊢ 𝐵 ∈ V |
14 | 13, 1 | ssexi 5250 | . . . . 5 ⊢ 𝐴 ∈ V |
15 | eqid 2740 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 8, 15 | ressplusg 16998 | . . . . 5 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
17 | 14, 16 | ax-mp 5 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐻) |
18 | 12, 17 | eqtri 2768 | . . 3 ⊢ ⨣ = (+g‘𝐻) |
19 | eqid 2740 | . . 3 ⊢ (+𝑓‘𝐻) = (+𝑓‘𝐻) | |
20 | 11, 18, 19 | plusffval 18330 | . 2 ⊢ (+𝑓‘𝐻) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦)) |
21 | 3, 7, 20 | 3eqtr4ri 2779 | 1 ⊢ (+𝑓‘𝐻) = ( ⨣ ↾ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 Vcvv 3431 ⊆ wss 3892 × cxp 5588 ↾ cres 5592 Fn wfn 6427 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 Basecbs 16910 ↾s cress 16939 +gcplusg 16960 +𝑓cplusf 18321 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-plusf 18323 |
This theorem is referenced by: xrge0pluscn 31886 xrge0tmdALT 31892 |
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