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Mirrors > Home > HSE Home > Th. List > hvaddcani | Structured version Visualization version GIF version |
Description: Cancellation law for vector addition. (Contributed by NM, 11-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvnegdi.1 | ⊢ 𝐴 ∈ ℋ |
hvnegdi.2 | ⊢ 𝐵 ∈ ℋ |
hvaddcan.3 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
hvaddcani | ⊢ ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ 𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7431 | . . 3 ⊢ ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) → ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐴)) = ((𝐴 +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐴))) | |
2 | hvnegdi.1 | . . . . 5 ⊢ 𝐴 ∈ ℋ | |
3 | hvnegdi.2 | . . . . 5 ⊢ 𝐵 ∈ ℋ | |
4 | neg1cn 12378 | . . . . . 6 ⊢ -1 ∈ ℂ | |
5 | 4, 2 | hvmulcli 30947 | . . . . 5 ⊢ (-1 ·ℎ 𝐴) ∈ ℋ |
6 | 2, 3, 5 | hvadd32i 30987 | . . . 4 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐴)) = ((𝐴 +ℎ (-1 ·ℎ 𝐴)) +ℎ 𝐵) |
7 | 2 | hvnegidi 30963 | . . . . 5 ⊢ (𝐴 +ℎ (-1 ·ℎ 𝐴)) = 0ℎ |
8 | 7 | oveq1i 7434 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐴)) +ℎ 𝐵) = (0ℎ +ℎ 𝐵) |
9 | 3 | hvaddlidi 30962 | . . . 4 ⊢ (0ℎ +ℎ 𝐵) = 𝐵 |
10 | 6, 8, 9 | 3eqtri 2758 | . . 3 ⊢ ((𝐴 +ℎ 𝐵) +ℎ (-1 ·ℎ 𝐴)) = 𝐵 |
11 | hvaddcan.3 | . . . . 5 ⊢ 𝐶 ∈ ℋ | |
12 | 2, 11, 5 | hvadd32i 30987 | . . . 4 ⊢ ((𝐴 +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐴)) = ((𝐴 +ℎ (-1 ·ℎ 𝐴)) +ℎ 𝐶) |
13 | 7 | oveq1i 7434 | . . . 4 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐴)) +ℎ 𝐶) = (0ℎ +ℎ 𝐶) |
14 | 11 | hvaddlidi 30962 | . . . 4 ⊢ (0ℎ +ℎ 𝐶) = 𝐶 |
15 | 12, 13, 14 | 3eqtri 2758 | . . 3 ⊢ ((𝐴 +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐴)) = 𝐶 |
16 | 1, 10, 15 | 3eqtr3g 2789 | . 2 ⊢ ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) → 𝐵 = 𝐶) |
17 | oveq2 7432 | . 2 ⊢ (𝐵 = 𝐶 → (𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶)) | |
18 | 16, 17 | impbii 208 | 1 ⊢ ((𝐴 +ℎ 𝐵) = (𝐴 +ℎ 𝐶) ↔ 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 (class class class)co 7424 1c1 11159 -cneg 11495 ℋchba 30852 +ℎ cva 30853 ·ℎ csm 30854 0ℎc0v 30857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-hvcom 30934 ax-hvass 30935 ax-hv0cl 30936 ax-hvaddid 30937 ax-hfvmul 30938 ax-hvmulid 30939 ax-hvdistr2 30942 ax-hvmul0 30943 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-ltxr 11303 df-sub 11496 df-neg 11497 df-hvsub 30904 |
This theorem is referenced by: hvsubaddi 30999 hvaddcan 31003 |
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