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Mirrors > Home > MPE Home > Th. List > vsfval | Structured version Visualization version GIF version |
Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vsfval.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
vsfval.3 | ⊢ 𝑀 = ( −𝑣 ‘𝑈) |
Ref | Expression |
---|---|
vsfval | ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-vs 28168 | . . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 ) | |
2 | 1 | fveq1i 6505 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈) |
3 | fo1st 7527 | . . . . . . . 8 ⊢ 1st :V–onto→V | |
4 | fof 6424 | . . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 1st :V⟶V |
6 | fco 6366 | . . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V) | |
7 | 5, 5, 6 | mp2an 680 | . . . . . 6 ⊢ (1st ∘ 1st ):V⟶V |
8 | df-va 28164 | . . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st ) | |
9 | 8 | feq1i 6340 | . . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V) |
10 | 7, 9 | mpbir 223 | . . . . 5 ⊢ +𝑣 :V⟶V |
11 | fvco3 6594 | . . . . 5 ⊢ (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) | |
12 | 10, 11 | mpan 678 | . . . 4 ⊢ (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
13 | 2, 12 | syl5eq 2828 | . . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
14 | 0ngrp 28080 | . . . . . 6 ⊢ ¬ ∅ ∈ GrpOp | |
15 | vex 3420 | . . . . . . . . . 10 ⊢ 𝑔 ∈ V | |
16 | 15 | rnex 7438 | . . . . . . . . 9 ⊢ ran 𝑔 ∈ V |
17 | 16, 16 | mpoex 7591 | . . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V |
18 | df-gdiv 28065 | . . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
19 | 17, 18 | dmmpti 6327 | . . . . . . 7 ⊢ dom /𝑔 = GrpOp |
20 | 19 | eleq2i 2859 | . . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp) |
21 | 14, 20 | mtbir 315 | . . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔 |
22 | ndmfv 6534 | . . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅) | |
23 | 21, 22 | mp1i 13 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅) |
24 | fvprc 6497 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
25 | 24 | fveq2d 6508 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅)) |
26 | fvprc 6497 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅) | |
27 | 23, 25, 26 | 3eqtr4rd 2827 | . . 3 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
28 | 13, 27 | pm2.61i 177 | . 2 ⊢ ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
29 | vsfval.3 | . 2 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
30 | vsfval.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
31 | 30 | fveq2i 6507 | . 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
32 | 28, 29, 31 | 3eqtr4i 2814 | 1 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1508 ∈ wcel 2051 Vcvv 3417 ∅c0 4181 dom cdm 5411 ran crn 5412 ∘ ccom 5415 ⟶wf 6189 –onto→wfo 6191 ‘cfv 6193 (class class class)co 6982 ∈ cmpo 6984 1st c1st 7505 GrpOpcgr 28058 invcgn 28060 /𝑔 cgs 28061 +𝑣 cpv 28154 −𝑣 cnsb 28158 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2752 ax-rep 5053 ax-sep 5064 ax-nul 5071 ax-pow 5123 ax-pr 5190 ax-un 7285 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2551 df-eu 2589 df-clab 2761 df-cleq 2773 df-clel 2848 df-nfc 2920 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3419 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-nul 4182 df-if 4354 df-pw 4427 df-sn 4445 df-pr 4447 df-op 4451 df-uni 4718 df-iun 4799 df-br 4935 df-opab 4997 df-mpt 5014 df-id 5316 df-xp 5417 df-rel 5418 df-cnv 5419 df-co 5420 df-dm 5421 df-rn 5422 df-res 5423 df-ima 5424 df-iota 6157 df-fun 6195 df-fn 6196 df-f 6197 df-f1 6198 df-fo 6199 df-f1o 6200 df-fv 6201 df-ov 6985 df-oprab 6986 df-mpo 6987 df-1st 7507 df-2nd 7508 df-grpo 28062 df-gdiv 28065 df-va 28164 df-vs 28168 |
This theorem is referenced by: nvm 28210 nvmfval 28213 nvnnncan1 28216 nvaddsub 28224 |
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