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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 31049 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | 3 | oveq2i 7442 | . 2 ⊢ (-1 ·ℎ (𝐴 −ℎ 𝐵)) = (-1 ·ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) |
5 | neg1cn 12378 | . . 3 ⊢ -1 ∈ ℂ | |
6 | 5, 2 | hvmulcli 31043 | . . 3 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
7 | 5, 1, 6 | hvdistr1i 31080 | . 2 ⊢ (-1 ·ℎ (𝐴 +ℎ (-1 ·ℎ 𝐵))) = ((-1 ·ℎ 𝐴) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐵))) |
8 | neg1mulneg1e1 12477 | . . . . . 6 ⊢ (-1 · -1) = 1 | |
9 | 8 | oveq1i 7441 | . . . . 5 ⊢ ((-1 · -1) ·ℎ 𝐵) = (1 ·ℎ 𝐵) |
10 | 5, 5, 2 | hvmulassi 31075 | . . . . 5 ⊢ ((-1 · -1) ·ℎ 𝐵) = (-1 ·ℎ (-1 ·ℎ 𝐵)) |
11 | ax-hvmulid 31035 | . . . . . 6 ⊢ (𝐵 ∈ ℋ → (1 ·ℎ 𝐵) = 𝐵) | |
12 | 2, 11 | ax-mp 5 | . . . . 5 ⊢ (1 ·ℎ 𝐵) = 𝐵 |
13 | 9, 10, 12 | 3eqtr3i 2771 | . . . 4 ⊢ (-1 ·ℎ (-1 ·ℎ 𝐵)) = 𝐵 |
14 | 13 | oveq1i 7441 | . . 3 ⊢ ((-1 ·ℎ (-1 ·ℎ 𝐵)) +ℎ (-1 ·ℎ 𝐴)) = (𝐵 +ℎ (-1 ·ℎ 𝐴)) |
15 | 5, 1 | hvmulcli 31043 | . . . 4 ⊢ (-1 ·ℎ 𝐴) ∈ ℋ |
16 | 5, 6 | hvmulcli 31043 | . . . 4 ⊢ (-1 ·ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
17 | 15, 16 | hvcomi 31048 | . . 3 ⊢ ((-1 ·ℎ 𝐴) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐵))) = ((-1 ·ℎ (-1 ·ℎ 𝐵)) +ℎ (-1 ·ℎ 𝐴)) |
18 | 2, 1 | hvsubvali 31049 | . . 3 ⊢ (𝐵 −ℎ 𝐴) = (𝐵 +ℎ (-1 ·ℎ 𝐴)) |
19 | 14, 17, 18 | 3eqtr4i 2773 | . 2 ⊢ ((-1 ·ℎ 𝐴) +ℎ (-1 ·ℎ (-1 ·ℎ 𝐵))) = (𝐵 −ℎ 𝐴) |
20 | 4, 7, 19 | 3eqtri 2767 | 1 ⊢ (-1 ·ℎ (𝐴 −ℎ 𝐵)) = (𝐵 −ℎ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 (class class class)co 7431 1c1 11154 · cmul 11158 -cneg 11491 ℋchba 30948 +ℎ cva 30949 ·ℎ csm 30950 −ℎ cmv 30954 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-hvcom 31030 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvmulass 31036 ax-hvdistr1 31037 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-hvsub 31000 |
This theorem is referenced by: hvnegdi 31096 hisubcomi 31133 normsubi 31170 normpar2i 31185 pjsslem 31708 pjcji 31713 |
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