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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 7251 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐵) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦))) | |
3 | 1, 1, 2 | mp2an 691 | . 2 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) ↾ (𝐴 × 𝐴)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦)) |
4 | ressplusf.4 | . . . 4 ⊢ ⨣ Fn (𝐵 × 𝐵) | |
5 | fnov 7261 | . . . 4 ⊢ ( ⨣ Fn (𝐵 × 𝐵) ↔ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦))) | |
6 | 4, 5 | mpbi 233 | . . 3 ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 ⨣ 𝑦)) |
7 | 6 | reseq1i 5814 | . 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 16547 | . . . 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 6659 | . . . . . 6 ⊢ 𝐵 ∈ V |
14 | 13, 1 | ssexi 5190 | . . . . 5 ⊢ 𝐴 ∈ V |
15 | eqid 2798 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 8, 15 | ressplusg 16604 | . . . . 5 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
17 | 14, 16 | ax-mp 5 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐻) |
18 | 12, 17 | eqtri 2821 | . . 3 ⊢ ⨣ = (+g‘𝐻) |
19 | eqid 2798 | . . 3 ⊢ (+𝑓‘𝐻) = (+𝑓‘𝐻) | |
20 | 11, 18, 19 | plusffval 17850 | . 2 ⊢ (+𝑓‘𝐻) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦)) |
21 | 3, 7, 20 | 3eqtr4ri 2832 | 1 ⊢ (+𝑓‘𝐻) = ( ⨣ ↾ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 × cxp 5517 ↾ cres 5521 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 Basecbs 16475 ↾s cress 16476 +gcplusg 16557 +𝑓cplusf 17841 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-plusf 17843 |
This theorem is referenced by: xrge0pluscn 31293 xrge0tmdALT 31299 |
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