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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 7540 | . . 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 7552 | . . . 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 5981 | . 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 17217 | . . . 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 6911 | . . . . . 6 ⊢ 𝐵 ∈ V |
14 | 13, 1 | ssexi 5322 | . . . . 5 ⊢ 𝐴 ∈ V |
15 | eqid 2728 | . . . . . 6 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 8, 15 | ressplusg 17270 | . . . . 5 ⊢ (𝐴 ∈ V → (+g‘𝐺) = (+g‘𝐻)) |
17 | 14, 16 | ax-mp 5 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐻) |
18 | 12, 17 | eqtri 2756 | . . 3 ⊢ ⨣ = (+g‘𝐻) |
19 | eqid 2728 | . . 3 ⊢ (+𝑓‘𝐻) = (+𝑓‘𝐻) | |
20 | 11, 18, 19 | plusffval 18605 | . 2 ⊢ (+𝑓‘𝐻) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐴 ↦ (𝑥 ⨣ 𝑦)) |
21 | 3, 7, 20 | 3eqtr4ri 2767 | 1 ⊢ (+𝑓‘𝐻) = ( ⨣ ↾ (𝐴 × 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⊆ wss 3947 × cxp 5676 ↾ cres 5680 Fn wfn 6543 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 Basecbs 17179 ↾s cress 17208 +gcplusg 17232 +𝑓cplusf 18596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-plusf 18598 |
This theorem is referenced by: xrge0pluscn 33541 xrge0tmdALT 33547 |
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