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Mirrors > Home > MPE Home > Th. List > Mathboxes > resvmulr | Structured version Visualization version GIF version |
Description: ·𝑠 is unaffected by scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) |
Ref | Expression |
---|---|
resvbas.1 | ⊢ 𝐻 = (𝐺 ↾v 𝐴) |
resvmulr.2 | ⊢ · = (.r‘𝐺) |
Ref | Expression |
---|---|
resvmulr | ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resvbas.1 | . 2 ⊢ 𝐻 = (𝐺 ↾v 𝐴) | |
2 | resvmulr.2 | . 2 ⊢ · = (.r‘𝐺) | |
3 | df-mulr 16571 | . 2 ⊢ .r = Slot 3 | |
4 | 3nn 11708 | . 2 ⊢ 3 ∈ ℕ | |
5 | 3re 11709 | . . 3 ⊢ 3 ∈ ℝ | |
6 | 3lt5 11807 | . . 3 ⊢ 3 < 5 | |
7 | 5, 6 | ltneii 10745 | . 2 ⊢ 3 ≠ 5 |
8 | 1, 2, 3, 4, 7 | resvlem 30897 | 1 ⊢ (𝐴 ∈ 𝑉 → · = (.r‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 3c3 11685 5c5 11687 .rcmulr 16558 ↾v cresv 30890 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-ndx 16478 df-slot 16479 df-sets 16482 df-mulr 16571 df-sca 16573 df-resv 30891 |
This theorem is referenced by: resv1r 30903 |
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