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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 28796 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | oveq2i 7166 | . 2 ⊢ (-1 ·ℎ (𝐴 −ℎ 𝐵)) = (-1 ·ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
5 | neg1cn 11750 | . . 3 ⊢ -1 ∈ ℂ | |
6 | 5, 2 | hvmulcli 28790 | . . 3 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
7 | 5, 1, 6 | hvdistr1i 28827 | . 2 ⊢ (-1 ·ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = ((-1 ·ℎ 𝐴) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐵))) |
8 | neg1mulneg1e1 11849 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
9 | 8 | oveq1i 7165 | . . . . 5 ⊢ ((-1 · -1) ·ℎ 𝐵) = (1 ·ℎ 𝐵) |
10 | 5, 5, 2 | hvmulassi 28822 | . . . . 5 ⊢ ((-1 · -1) ·ℎ 𝐵) = (-1 ·ℎ (-1 ·ℎ 𝐵)) |
11 | ax-hvmulid 28782 | . . . . . 6 ⊢ (𝐵 ∈ ℋ → (1 ·ℎ 𝐵) = 𝐵) | |
12 | 2, 11 | ax-mp 5 | . . . . 5 ⊢ (1 ·ℎ 𝐵) = 𝐵 |
13 | 9, 10, 12 | 3eqtr3i 2852 | . . . 4 ⊢ (-1 ·ℎ (-1 ·ℎ 𝐵)) = 𝐵 |
14 | 13 | oveq1i 7165 | . . 3 ⊢ ((-1 ·ℎ (-1 ·ℎ 𝐵)) +ℎ (-1 ·ℎ 𝐴)) = (𝐵 +ℎ (-1 ·ℎ 𝐴)) |
15 | 5, 1 | hvmulcli 28790 | . . . 4 ⊢ (-1 ·ℎ 𝐴) ∈ ℋ |
16 | 5, 6 | hvmulcli 28790 | . . . 4 ⊢ (-1 ·ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
17 | 15, 16 | hvcomi 28795 | . . 3 ⊢ ((-1 ·ℎ 𝐴) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐵))) = ((-1 ·ℎ (-1 ·ℎ 𝐵)) +ℎ (-1 ·ℎ 𝐴)) |
18 | 2, 1 | hvsubvali 28796 | . . 3 ⊢ (𝐵 −ℎ 𝐴) = (𝐵 +ℎ (-1 ·ℎ 𝐴)) |
19 | 14, 17, 18 | 3eqtr4i 2854 | . 2 ⊢ ((-1 ·ℎ 𝐴) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐵))) = (𝐵 −ℎ 𝐴) |
20 | 4, 7, 19 | 3eqtri 2848 | 1 ⊢ (-1 ·ℎ (𝐴 −ℎ 𝐵)) = (𝐵 −ℎ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 (class class class)co 7155 1c1 10537 · cmul 10541 -cneg 10870 ℋchba 28695 +ℎ cva 28696 ·ℎ csm 28697 −ℎ cmv 28701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-hvcom 28777 ax-hfvmul 28781 ax-hvmulid 28782 ax-hvmulass 28783 ax-hvdistr1 28784 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-ltxr 10679 df-sub 10871 df-neg 10872 df-hvsub 28747 |
This theorem is referenced by: hvnegdi 28843 hisubcomi 28880 normsubi 28917 normpar2i 28932 pjsslem 29455 pjcji 29460 |
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