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Mirrors > Home > HSE Home > Th. List > hvnegdii | Structured version Visualization version GIF version |
Description: Distribution of negative over subtraction. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvnegdi.1 | ⊢ 𝐴 ∈ ℋ |
hvnegdi.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
hvnegdii | ⊢ (-1 ·ℎ (𝐴 −ℎ 𝐵)) = (𝐵 −ℎ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvnegdi.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
2 | hvnegdi.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 2 | hvsubvali 28803 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | oveq2i 7146 | . 2 ⊢ (-1 ·ℎ (𝐴 −ℎ 𝐵)) = (-1 ·ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
5 | neg1cn 11739 | . . 3 ⊢ -1 ∈ ℂ | |
6 | 5, 2 | hvmulcli 28797 | . . 3 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
7 | 5, 1, 6 | hvdistr1i 28834 | . 2 ⊢ (-1 ·ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = ((-1 ·ℎ 𝐴) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐵))) |
8 | neg1mulneg1e1 11838 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
9 | 8 | oveq1i 7145 | . . . . 5 ⊢ ((-1 · -1) ·ℎ 𝐵) = (1 ·ℎ 𝐵) |
10 | 5, 5, 2 | hvmulassi 28829 | . . . . 5 ⊢ ((-1 · -1) ·ℎ 𝐵) = (-1 ·ℎ (-1 ·ℎ 𝐵)) |
11 | ax-hvmulid 28789 | . . . . . 6 ⊢ (𝐵 ∈ ℋ → (1 ·ℎ 𝐵) = 𝐵) | |
12 | 2, 11 | ax-mp 5 | . . . . 5 ⊢ (1 ·ℎ 𝐵) = 𝐵 |
13 | 9, 10, 12 | 3eqtr3i 2829 | . . . 4 ⊢ (-1 ·ℎ (-1 ·ℎ 𝐵)) = 𝐵 |
14 | 13 | oveq1i 7145 | . . 3 ⊢ ((-1 ·ℎ (-1 ·ℎ 𝐵)) +ℎ (-1 ·ℎ 𝐴)) = (𝐵 +ℎ (-1 ·ℎ 𝐴)) |
15 | 5, 1 | hvmulcli 28797 | . . . 4 ⊢ (-1 ·ℎ 𝐴) ∈ ℋ |
16 | 5, 6 | hvmulcli 28797 | . . . 4 ⊢ (-1 ·ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
17 | 15, 16 | hvcomi 28802 | . . 3 ⊢ ((-1 ·ℎ 𝐴) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐵))) = ((-1 ·ℎ (-1 ·ℎ 𝐵)) +ℎ (-1 ·ℎ 𝐴)) |
18 | 2, 1 | hvsubvali 28803 | . . 3 ⊢ (𝐵 −ℎ 𝐴) = (𝐵 +ℎ (-1 ·ℎ 𝐴)) |
19 | 14, 17, 18 | 3eqtr4i 2831 | . 2 ⊢ ((-1 ·ℎ 𝐴) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐵))) = (𝐵 −ℎ 𝐴) |
20 | 4, 7, 19 | 3eqtri 2825 | 1 ⊢ (-1 ·ℎ (𝐴 −ℎ 𝐵)) = (𝐵 −ℎ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 1c1 10527 · cmul 10531 -cneg 10860 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 −ℎ cmv 28708 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-hvcom 28784 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr1 28791 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 df-hvsub 28754 |
This theorem is referenced by: hvnegdi 28850 hisubcomi 28887 normsubi 28924 normpar2i 28939 pjsslem 29462 pjcji 29467 |
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