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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 12322 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | hvmulcl 30253 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ 𝐵) ∈ ℋ) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·ℎ 𝐵) ∈ ℋ) |
4 | hvaddsubass 30281 | . . 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 30256 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) | |
7 | 6 | 3adant3 1132 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
8 | 7 | oveq1d 7420 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) −ℎ 𝐶) = ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ 𝐶)) |
9 | simp1 1136 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐴 ∈ ℋ) | |
10 | hvaddcl 30252 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) ∈ ℋ) | |
11 | 10 | 3adant1 1130 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) ∈ ℋ) |
12 | hvsubval 30256 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +ℎ 𝐶) ∈ ℋ) → (𝐴 −ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (-1 ·ℎ (𝐵 +ℎ 𝐶)))) | |
13 | 9, 11, 12 | syl2anc 584 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (-1 ·ℎ (𝐵 +ℎ 𝐶)))) |
14 | hvsubval 30256 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) | |
15 | 3, 14 | sylan 580 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
16 | 15 | 3adant1 1130 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
17 | ax-hvdistr1 30248 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ (𝐵 +ℎ 𝐶)) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) | |
18 | 1, 17 | mp3an1 1448 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ (𝐵 +ℎ 𝐶)) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
19 | 18 | 3adant1 1130 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ (𝐵 +ℎ 𝐶)) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
20 | 16, 19 | eqtr4d 2775 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = (-1 ·ℎ (𝐵 +ℎ 𝐶))) |
21 | 20 | oveq2d 7421 | . . 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 2775 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ ((-1 ·ℎ 𝐵) −ℎ 𝐶))) |
23 | 5, 8, 22 | 3eqtr4d 2782 | 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 1087 = wceq 1541 ∈ wcel 2106 (class class class)co 7405 ℂcc 11104 1c1 11107 -cneg 11441 ℋchba 30159 +ℎ cva 30160 ·ℎ csm 30161 −ℎ cmv 30165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-hfvadd 30240 ax-hvass 30242 ax-hfvmul 30245 ax-hvdistr1 30248 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-sub 11442 df-neg 11443 df-hvsub 30211 |
This theorem is referenced by: hvsub32 30285 hvsubassi 30295 pjhthlem1 30631 |
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