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Mirrors > Home > HSE Home > Th. List > honegsubi | Structured version Visualization version GIF version |
Description: Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hodseq.2 | ⊢ 𝑆: ℋ⟶ ℋ |
hodseq.3 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
honegsubi | ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hodseq.2 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | neg1cn 12378 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
3 | hodseq.3 | . . . . . . 7 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | homulcl 31788 | . . . . . . 7 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (-1 ·op 𝑇): ℋ⟶ ℋ) | |
5 | 2, 3, 4 | mp2an 692 | . . . . . 6 ⊢ (-1 ·op 𝑇): ℋ⟶ ℋ |
6 | hosval 31769 | . . . . . 6 ⊢ ((𝑆: ℋ⟶ ℋ ∧ (-1 ·op 𝑇): ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) | |
7 | 1, 5, 6 | mp3an12 1450 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) |
8 | 1 | ffvelcdmi 7103 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
9 | 3 | ffvelcdmi 7103 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
10 | hvsubval 31045 | . . . . . . 7 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) | |
11 | 8, 9, 10 | syl2anc 584 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) |
12 | homval 31770 | . . . . . . . 8 ⊢ ((-1 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) | |
13 | 2, 3, 12 | mp3an12 1450 | . . . . . . 7 ⊢ (𝑥 ∈ ℋ → ((-1 ·op 𝑇)‘𝑥) = (-1 ·ℎ (𝑇‘𝑥))) |
14 | 13 | oveq2d 7447 | . . . . . 6 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥)) = ((𝑆‘𝑥) +ℎ (-1 ·ℎ (𝑇‘𝑥)))) |
15 | 11, 14 | eqtr4d 2778 | . . . . 5 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) −ℎ (𝑇‘𝑥)) = ((𝑆‘𝑥) +ℎ ((-1 ·op 𝑇)‘𝑥))) |
16 | 7, 15 | eqtr4d 2778 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
17 | hodval 31771 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) | |
18 | 1, 3, 17 | mp3an12 1450 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 −op 𝑇)‘𝑥) = ((𝑆‘𝑥) −ℎ (𝑇‘𝑥))) |
19 | 16, 18 | eqtr4d 2778 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥)) |
20 | 19 | rgen 3061 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥) |
21 | 1, 5 | hoaddcli 31797 | . . 3 ⊢ (𝑆 +op (-1 ·op 𝑇)): ℋ⟶ ℋ |
22 | 1, 3 | hosubcli 31798 | . . 3 ⊢ (𝑆 −op 𝑇): ℋ⟶ ℋ |
23 | 21, 22 | hoeqi 31790 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op (-1 ·op 𝑇))‘𝑥) = ((𝑆 −op 𝑇)‘𝑥) ↔ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇)) |
24 | 20, 23 | mpbi 230 | 1 ⊢ (𝑆 +op (-1 ·op 𝑇)) = (𝑆 −op 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ∀wral 3059 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 1c1 11154 -cneg 11491 ℋchba 30948 +ℎ cva 30949 ·ℎ csm 30950 −ℎ cmv 30954 +op chos 30967 ·op chot 30968 −op chod 30969 |
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-rep 5285 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-hilex 31028 ax-hfvadd 31029 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-map 8867 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 df-hosum 31759 df-homul 31760 df-hodif 31761 |
This theorem is referenced by: honegsub 31828 hosubeq0i 31855 lnophdi 32031 bdophdi 32126 nmoptri2i 32128 |
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