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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 11754 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | hvmulcl 28792 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (-1 ·ℎ 𝐵) ∈ ℋ) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐵 ∈ ℋ → (-1 ·ℎ 𝐵) ∈ ℋ) |
4 | hvaddsubass 28820 | . . 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 1160 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ 𝐶) = (𝐴 +ℎ ((-1 ·ℎ 𝐵) −ℎ 𝐶))) |
6 | hvsubval 28795 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) | |
7 | 6 | 3adant3 1128 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
8 | 7 | oveq1d 7173 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 −ℎ 𝐵) −ℎ 𝐶) = ((𝐴 +ℎ (-1 ·ℎ 𝐵)) −ℎ 𝐶)) |
9 | simp1 1132 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → 𝐴 ∈ ℋ) | |
10 | hvaddcl 28791 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) ∈ ℋ) | |
11 | 10 | 3adant1 1126 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ 𝐶) ∈ ℋ) |
12 | hvsubval 28795 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ (𝐵 +ℎ 𝐶) ∈ ℋ) → (𝐴 −ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (-1 ·ℎ (𝐵 +ℎ 𝐶)))) | |
13 | 9, 11, 12 | syl2anc 586 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ (-1 ·ℎ (𝐵 +ℎ 𝐶)))) |
14 | hvsubval 28795 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐵) ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) | |
15 | 3, 14 | sylan 582 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
16 | 15 | 3adant1 1126 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
17 | ax-hvdistr1 28787 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ (𝐵 +ℎ 𝐶)) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) | |
18 | 1, 17 | mp3an1 1444 | . . . . . 6 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ (𝐵 +ℎ 𝐶)) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
19 | 18 | 3adant1 1126 | . . . . 5 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ (𝐵 +ℎ 𝐶)) = ((-1 ·ℎ 𝐵) +ℎ (-1 ·ℎ 𝐶))) |
20 | 16, 19 | eqtr4d 2861 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((-1 ·ℎ 𝐵) −ℎ 𝐶) = (-1 ·ℎ (𝐵 +ℎ 𝐶))) |
21 | 20 | oveq2d 7174 | . . 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 2861 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ (𝐵 +ℎ 𝐶)) = (𝐴 +ℎ ((-1 ·ℎ 𝐵) −ℎ 𝐶))) |
23 | 5, 8, 22 | 3eqtr4d 2868 | 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 1083 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 ℂcc 10537 1c1 10540 -cneg 10873 ℋchba 28698 +ℎ cva 28699 ·ℎ csm 28700 −ℎ cmv 28704 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-hfvadd 28779 ax-hvass 28781 ax-hfvmul 28784 ax-hvdistr1 28787 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-ltxr 10682 df-sub 10874 df-neg 10875 df-hvsub 28750 |
This theorem is referenced by: hvsub32 28824 hvsubassi 28834 pjhthlem1 29170 |
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