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Mirrors > Home > MPE Home > Th. List > resssca | Structured version Visualization version GIF version |
Description: Scalar is unaffected by restriction. (Contributed by Mario Carneiro, 7-Dec-2014.) |
Ref | Expression |
---|---|
resssca.1 | ⊢ 𝐻 = (𝐺 ↾s 𝐴) |
resssca.2 | ⊢ 𝐹 = (Scalar‘𝐺) |
Ref | Expression |
---|---|
resssca | ⊢ (𝐴 ∈ 𝑉 → 𝐹 = (Scalar‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resssca.1 | . 2 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
2 | resssca.2 | . 2 ⊢ 𝐹 = (Scalar‘𝐺) | |
3 | df-sca 16581 | . 2 ⊢ Scalar = Slot 5 | |
4 | 5nn 11724 | . 2 ⊢ 5 ∈ ℕ | |
5 | 1lt5 11818 | . 2 ⊢ 1 < 5 | |
6 | 1, 2, 3, 4, 5 | resslem 16557 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐹 = (Scalar‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 5c5 11696 ↾s cress 16484 Scalarcsca 16568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-sca 16581 |
This theorem is referenced by: islss3 19731 reslmhm 19824 reslmhm2 19825 reslmhm2b 19826 pj1lmhm 19872 lsslvec 19879 issubassa3 20097 ressascl 20125 mplsca 20225 ply1sca 20421 phlssphl 20803 frlmsca 20897 lsslindf 20974 scmatghm 21142 lssnlm 23310 lssnvc 23311 cphsscph 23854 lssbn 23955 cmslssbn 23975 csschl 23979 rrxsca 23999 xrge0slmod 30917 fedgmullem2 31026 sitmcl 31609 repwsmet 35127 rrnequiv 35128 lcdsca 38750 |
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