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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 27454 | . . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 ) | |
2 | 1 | fveq1i 6192 | . . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈) |
3 | fo1st 7188 | . . . . . . . 8 ⊢ 1st :V–onto→V | |
4 | fof 6115 | . . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ 1st :V⟶V |
6 | fco 6058 | . . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V) | |
7 | 5, 5, 6 | mp2an 708 | . . . . . 6 ⊢ (1st ∘ 1st ):V⟶V |
8 | df-va 27450 | . . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st ) | |
9 | 8 | feq1i 6036 | . . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V) |
10 | 7, 9 | mpbir 221 | . . . . 5 ⊢ +𝑣 :V⟶V |
11 | fvco3 6275 | . . . . 5 ⊢ (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) | |
12 | 10, 11 | mpan 706 | . . . 4 ⊢ (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
13 | 2, 12 | syl5eq 2668 | . . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
14 | 0ngrp 27365 | . . . . . 6 ⊢ ¬ ∅ ∈ GrpOp | |
15 | vex 3203 | . . . . . . . . . 10 ⊢ 𝑔 ∈ V | |
16 | 15 | rnex 7100 | . . . . . . . . 9 ⊢ ran 𝑔 ∈ V |
17 | 16, 16 | mpt2ex 7247 | . . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V |
18 | df-gdiv 27350 | . . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦)))) | |
19 | 17, 18 | dmmpti 6023 | . . . . . . 7 ⊢ dom /𝑔 = GrpOp |
20 | 19 | eleq2i 2693 | . . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp) |
21 | 14, 20 | mtbir 313 | . . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔 |
22 | ndmfv 6218 | . . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅) | |
23 | 21, 22 | mp1i 13 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅) |
24 | fvprc 6185 | . . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅) | |
25 | 24 | fveq2d 6195 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅)) |
26 | fvprc 6185 | . . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅) | |
27 | 23, 25, 26 | 3eqtr4rd 2667 | . . 3 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))) |
28 | 13, 27 | pm2.61i 176 | . 2 ⊢ ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
29 | vsfval.3 | . 2 ⊢ 𝑀 = ( −𝑣 ‘𝑈) | |
30 | vsfval.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
31 | 30 | fveq2i 6194 | . 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈)) |
32 | 28, 29, 31 | 3eqtr4i 2654 | 1 ⊢ 𝑀 = ( /𝑔 ‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∅c0 3915 dom cdm 5114 ran crn 5115 ∘ ccom 5118 ⟶wf 5884 –onto→wfo 5886 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1st c1st 7166 GrpOpcgr 27343 invcgn 27345 /𝑔 cgs 27346 +𝑣 cpv 27440 −𝑣 cnsb 27444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-grpo 27347 df-gdiv 27350 df-va 27450 df-vs 27454 |
This theorem is referenced by: nvm 27496 nvmfval 27499 nvnnncan1 27502 nvaddsub 27510 |
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