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Mirrors > Home > HSE Home > Th. List > h2hsm | Structured version Visualization version Unicode version |
Description: The scalar product operation of Hilbert space. (Contributed by NM, 31-May-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
h2h.1 | |
h2h.2 |
Ref | Expression |
---|---|
h2hsm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . . 4 | |
2 | 1 | smfval 27460 | . . 3 |
3 | opex 4932 | . . . . 5 | |
4 | h2h.1 | . . . . . . . 8 | |
5 | h2h.2 | . . . . . . . 8 | |
6 | 4, 5 | eqeltrri 2698 | . . . . . . 7 |
7 | nvex 27466 | . . . . . . 7 | |
8 | 6, 7 | ax-mp 5 | . . . . . 6 |
9 | 8 | simp3i 1072 | . . . . 5 |
10 | 3, 9 | op1st 7176 | . . . 4 |
11 | 10 | fveq2i 6194 | . . 3 |
12 | 8 | simp1i 1070 | . . . 4 |
13 | 8 | simp2i 1071 | . . . 4 |
14 | 12, 13 | op2nd 7177 | . . 3 |
15 | 2, 11, 14 | 3eqtrri 2649 | . 2 |
16 | 4 | fveq2i 6194 | . 2 |
17 | 15, 16 | eqtr4i 2647 | 1 |
Colors of variables: wff setvar class |
Syntax hints: w3a 1037 wceq 1483 wcel 1990 cvv 3200 cop 4183 cfv 5888 c1st 7166 c2nd 7167 cnv 27439 cns 27442 cva 27777 csm 27778 cno 27780 |
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-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-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 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-fo 5894 df-fv 5896 df-oprab 6654 df-1st 7168 df-2nd 7169 df-vc 27414 df-nv 27447 df-sm 27452 |
This theorem is referenced by: h2hvs 27834 axhfvmul-zf 27844 axhvmulid-zf 27845 axhvmulass-zf 27846 axhvdistr1-zf 27847 axhvdistr2-zf 27848 axhvmul0-zf 27849 axhis3-zf 27853 hhsm 28026 |
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