![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > hvsubass | Structured version Visualization version GIF version |
Description: Hilbert vector space associative law for subtraction. (Contributed by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvsubass | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) −ℎ 𝐶) = (𝐴 −ℎ (𝐵 +ℎ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 11501 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | hvmulcl 28459 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ 𝐵) ∈ ℋ) | |
3 | 1, 2 | mpan 680 | . . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·ℎ 𝐵) ∈ ℋ) |
4 | hvaddsubass 28487 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ (-1 ·ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ 𝐶) = (𝐴 +ℎ ((-1 ·ℎ 𝐵) −ℎ 𝐶))) | |
5 | 3, 4 | syl3an2 1164 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ 𝐶) = (𝐴 +ℎ ((-1 ·ℎ 𝐵) −ℎ 𝐶))) |
6 | hvsubval 28462 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) | |
7 | 6 | 3adant3 1123 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
8 | 7 | oveq1d 6939 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) −ℎ 𝐶) = ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ 𝐶)) |
9 | simp1 1127 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐴 ∈ ℋ) | |
10 | hvaddcl 28458 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) ∈ ℋ) | |
11 | 10 | 3adant1 1121 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) ∈ ℋ) |
12 | hvsubval 28462 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +ℎ 𝐶) ∈ ℋ) → (𝐴 −ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (-1 ·ℎ (𝐵 +ℎ 𝐶)))) | |
13 | 9, 11, 12 | syl2anc 579 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (-1 ·ℎ (𝐵 +ℎ 𝐶)))) |
14 | hvsubval 28462 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) | |
15 | 3, 14 | sylan 575 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
16 | 15 | 3adant1 1121 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
17 | ax-hvdistr1 28454 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ (𝐵 +ℎ 𝐶)) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) | |
18 | 1, 17 | mp3an1 1521 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ (𝐵 +ℎ 𝐶)) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
19 | 18 | 3adant1 1121 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ (𝐵 +ℎ 𝐶)) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
20 | 16, 19 | eqtr4d 2817 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = (-1 ·ℎ (𝐵 +ℎ 𝐶))) |
21 | 20 | oveq2d 6940 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ ((-1 ·ℎ 𝐵) −ℎ 𝐶)) = (𝐴 +ℎ (-1 ·ℎ (𝐵 +ℎ 𝐶)))) |
22 | 13, 21 | eqtr4d 2817 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ ((-1 ·ℎ 𝐵) −ℎ 𝐶))) |
23 | 5, 8, 22 | 3eqtr4d 2824 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) −ℎ 𝐶) = (𝐴 −ℎ (𝐵 +ℎ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 (class class class)co 6924 ℂcc 10272 1c1 10275 -cneg 10609 ℋchba 28365 +ℎ cva 28366 ·ℎ csm 28367 −ℎ cmv 28371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-hfvadd 28446 ax-hvass 28448 ax-hfvmul 28451 ax-hvdistr1 28454 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-po 5276 df-so 5277 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-ltxr 10418 df-sub 10610 df-neg 10611 df-hvsub 28417 |
This theorem is referenced by: hvsub32 28491 hvsubassi 28501 pjhthlem1 28839 |
Copyright terms: Public domain | W3C validator |