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Mirrors > Home > HSE Home > Th. List > hvaddsub12 | Structured version Visualization version GIF version |
Description: Commutative/associative law. (Contributed by NM, 19-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvaddsub12 | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐶)) = (𝐵 +ℎ (𝐴 −ℎ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 12378 | . . . 4 ⊢ -1 ∈ ℂ | |
2 | hvmulcl 31042 | . . . 4 ⊢ ((-1 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (-1 ·ℎ 𝐶) ∈ ℋ) | |
3 | 1, 2 | mpan 690 | . . 3 ⊢ (𝐶 ∈ ℋ → (-1 ·ℎ 𝐶) ∈ ℋ) |
4 | hvadd12 31064 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ (-1 ·ℎ 𝐶) ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐶))) = (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐶)))) | |
5 | 3, 4 | syl3an3 1164 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐶))) = (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐶)))) |
6 | hvsubval 31045 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 −ℎ 𝐶) = (𝐵 +ℎ (-1 ·ℎ 𝐶))) | |
7 | 6 | oveq2d 7447 | . . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐶)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐶)))) |
8 | 7 | 3adant1 1129 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐶)) = (𝐴 +ℎ (𝐵 +ℎ (-1 ·ℎ 𝐶)))) |
9 | hvsubval 31045 | . . . 4 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶))) | |
10 | 9 | oveq2d 7447 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ (𝐴 −ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐶)))) |
11 | 10 | 3adant2 1130 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 +ℎ (𝐴 −ℎ 𝐶)) = (𝐵 +ℎ (𝐴 +ℎ (-1 ·ℎ 𝐶)))) |
12 | 5, 8, 11 | 3eqtr4d 2785 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 +ℎ (𝐵 −ℎ 𝐶)) = (𝐵 +ℎ (𝐴 −ℎ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 (class class class)co 7431 ℂcc 11151 1c1 11154 -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-hvass 31031 ax-hfvmul 31034 |
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: 5oalem1 31683 3oalem2 31692 pjcji 31713 pjclem4 32228 pj3si 32236 |
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