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Mirrors > Home > HSE Home > Th. List > hosubeq0i | Structured version Visualization version GIF version |
Description: If the difference between two operators is zero, they are equal. (Contributed by NM, 27-Jul-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hosd1.2 | ⊢ 𝑇: ℋ⟶ ℋ |
hosd1.3 | ⊢ 𝑈: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hosubeq0i | ⊢ ((𝑇 −op 𝑈) = 0hop ↔ 𝑇 = 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hosd1.2 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ | |
2 | hosd1.3 | . . . . . 6 ⊢ 𝑈: ℋ⟶ ℋ | |
3 | 1, 2 | honegsubi 29567 | . . . . 5 ⊢ (𝑇 +op (-1 ·op 𝑈)) = (𝑇 −op 𝑈) |
4 | 3 | eqeq1i 2826 | . . . 4 ⊢ ((𝑇 +op (-1 ·op 𝑈)) = 0hop ↔ (𝑇 −op 𝑈) = 0hop ) |
5 | oveq1 7157 | . . . 4 ⊢ ((𝑇 +op (-1 ·op 𝑈)) = 0hop → ((𝑇 +op (-1 ·op 𝑈)) +op 𝑈) = ( 0hop +op 𝑈)) | |
6 | 4, 5 | sylbir 237 | . . 3 ⊢ ((𝑇 −op 𝑈) = 0hop → ((𝑇 +op (-1 ·op 𝑈)) +op 𝑈) = ( 0hop +op 𝑈)) |
7 | neg1cn 11745 | . . . . . 6 ⊢ -1 ∈ ℂ | |
8 | homulcl 29530 | . . . . . 6 ⊢ ((-1 ∈ ℂ ∧ 𝑈: ℋ⟶ ℋ) → (-1 ·op 𝑈): ℋ⟶ ℋ) | |
9 | 7, 2, 8 | mp2an 690 | . . . . 5 ⊢ (-1 ·op 𝑈): ℋ⟶ ℋ |
10 | 1, 9, 2 | hoadd32i 29549 | . . . 4 ⊢ ((𝑇 +op (-1 ·op 𝑈)) +op 𝑈) = ((𝑇 +op 𝑈) +op (-1 ·op 𝑈)) |
11 | 1, 2, 9 | hoaddassi 29547 | . . . . 5 ⊢ ((𝑇 +op 𝑈) +op (-1 ·op 𝑈)) = (𝑇 +op (𝑈 +op (-1 ·op 𝑈))) |
12 | 2, 2 | honegsubi 29567 | . . . . . . . 8 ⊢ (𝑈 +op (-1 ·op 𝑈)) = (𝑈 −op 𝑈) |
13 | 2 | hodidi 29558 | . . . . . . . 8 ⊢ (𝑈 −op 𝑈) = 0hop |
14 | 12, 13 | eqtri 2844 | . . . . . . 7 ⊢ (𝑈 +op (-1 ·op 𝑈)) = 0hop |
15 | 14 | oveq2i 7161 | . . . . . 6 ⊢ (𝑇 +op (𝑈 +op (-1 ·op 𝑈))) = (𝑇 +op 0hop ) |
16 | 1 | hoaddid1i 29557 | . . . . . 6 ⊢ (𝑇 +op 0hop ) = 𝑇 |
17 | 15, 16 | eqtri 2844 | . . . . 5 ⊢ (𝑇 +op (𝑈 +op (-1 ·op 𝑈))) = 𝑇 |
18 | 11, 17 | eqtri 2844 | . . . 4 ⊢ ((𝑇 +op 𝑈) +op (-1 ·op 𝑈)) = 𝑇 |
19 | 10, 18 | eqtri 2844 | . . 3 ⊢ ((𝑇 +op (-1 ·op 𝑈)) +op 𝑈) = 𝑇 |
20 | ho0f 29522 | . . . . 5 ⊢ 0hop : ℋ⟶ ℋ | |
21 | 20, 2 | hoaddcomi 29543 | . . . 4 ⊢ ( 0hop +op 𝑈) = (𝑈 +op 0hop ) |
22 | 2 | hoaddid1i 29557 | . . . 4 ⊢ (𝑈 +op 0hop ) = 𝑈 |
23 | 21, 22 | eqtri 2844 | . . 3 ⊢ ( 0hop +op 𝑈) = 𝑈 |
24 | 6, 19, 23 | 3eqtr3g 2879 | . 2 ⊢ ((𝑇 −op 𝑈) = 0hop → 𝑇 = 𝑈) |
25 | oveq1 7157 | . . 3 ⊢ (𝑇 = 𝑈 → (𝑇 −op 𝑈) = (𝑈 −op 𝑈)) | |
26 | 25, 13 | syl6eq 2872 | . 2 ⊢ (𝑇 = 𝑈 → (𝑇 −op 𝑈) = 0hop ) |
27 | 24, 26 | impbii 211 | 1 ⊢ ((𝑇 −op 𝑈) = 0hop ↔ 𝑇 = 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1533 ∈ wcel 2110 ⟶wf 6346 (class class class)co 7150 ℂcc 10529 1c1 10532 -cneg 10865 ℋchba 28690 +op chos 28709 ·op chot 28710 −op chod 28711 0hop ch0o 28714 |
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 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cc 9851 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 ax-hilex 28770 ax-hfvadd 28771 ax-hvcom 28772 ax-hvass 28773 ax-hv0cl 28774 ax-hvaddid 28775 ax-hfvmul 28776 ax-hvmulid 28777 ax-hvmulass 28778 ax-hvdistr1 28779 ax-hvdistr2 28780 ax-hvmul0 28781 ax-hfi 28850 ax-his1 28853 ax-his2 28854 ax-his3 28855 ax-his4 28856 ax-hcompl 28973 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 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-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-cn 21829 df-cnp 21830 df-lm 21831 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cfil 23852 df-cau 23853 df-cmet 23854 df-grpo 28264 df-gid 28265 df-ginv 28266 df-gdiv 28267 df-ablo 28316 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-0v 28369 df-vs 28370 df-nmcv 28371 df-ims 28372 df-dip 28472 df-ssp 28493 df-ph 28584 df-cbn 28634 df-hnorm 28739 df-hba 28740 df-hvsub 28742 df-hlim 28743 df-hcau 28744 df-sh 28978 df-ch 28992 df-oc 29023 df-ch0 29024 df-shs 29079 df-pjh 29166 df-hosum 29501 df-homul 29502 df-hodif 29503 df-h0op 29519 |
This theorem is referenced by: lnopeqi 29779 |
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