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Mirrors > Home > MPE Home > Th. List > tngip | Structured version Visualization version GIF version |
Description: The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
tngbas.t | ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) |
tngip.2 | ⊢ , = (·𝑖‘𝐺) |
Ref | Expression |
---|---|
tngip | ⊢ (𝑁 ∈ 𝑉 → , = (·𝑖‘𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngip.2 | . 2 ⊢ , = (·𝑖‘𝐺) | |
2 | tngbas.t | . . 3 ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) | |
3 | df-ip 16699 | . . 3 ⊢ ·𝑖 = Slot 8 | |
4 | 8nn 11824 | . . 3 ⊢ 8 ∈ ℕ | |
5 | 8lt9 11928 | . . 3 ⊢ 8 < 9 | |
6 | 2, 3, 4, 5 | tnglem 23406 | . 2 ⊢ (𝑁 ∈ 𝑉 → (·𝑖‘𝐺) = (·𝑖‘𝑇)) |
7 | 1, 6 | syl5eq 2786 | 1 ⊢ (𝑁 ∈ 𝑉 → , = (·𝑖‘𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6350 (class class class)co 7183 8c8 11790 ·𝑖cip 16686 toNrmGrp ctng 23344 |
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 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7492 ax-cnex 10684 ax-resscn 10685 ax-1cn 10686 ax-icn 10687 ax-addcl 10688 ax-addrcl 10689 ax-mulcl 10690 ax-mulrcl 10691 ax-mulcom 10692 ax-addass 10693 ax-mulass 10694 ax-distr 10695 ax-i2m1 10696 ax-1ne0 10697 ax-1rid 10698 ax-rnegex 10699 ax-rrecex 10700 ax-cnre 10701 ax-pre-lttri 10702 ax-pre-lttrn 10703 ax-pre-ltadd 10704 ax-pre-mulgt0 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6186 df-on 6187 df-lim 6188 df-suc 6189 df-iota 6308 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7140 df-ov 7186 df-oprab 7187 df-mpo 7188 df-om 7613 df-wrecs 7989 df-recs 8050 df-rdg 8088 df-er 8333 df-en 8569 df-dom 8570 df-sdom 8571 df-pnf 10768 df-mnf 10769 df-xr 10770 df-ltxr 10771 df-le 10772 df-sub 10963 df-neg 10964 df-nn 11730 df-2 11792 df-3 11793 df-4 11794 df-5 11795 df-6 11796 df-7 11797 df-8 11798 df-9 11799 df-n0 11990 df-z 12076 df-dec 12193 df-ndx 16602 df-slot 16603 df-sets 16606 df-ip 16699 df-tset 16700 df-ds 16703 df-tng 23350 |
This theorem is referenced by: tcphip 23990 |
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