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Mirrors > Home > HSE Home > Th. List > h1de2ctlem | Structured version Visualization version GIF version |
Description: Lemma for h1de2ci 31386. (Contributed by NM, 19-Jul-2001.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
h1de2.1 | ⊢ 𝐴 ∈ ℋ |
h1de2.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
h1de2ctlem | ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | h1de2.1 | . . . . . . . 8 ⊢ 𝐴 ∈ ℋ | |
2 | 1 | elexi 3493 | . . . . . . 7 ⊢ 𝐴 ∈ V |
3 | 2 | elsn 4647 | . . . . . 6 ⊢ (𝐴 ∈ {0ℎ} ↔ 𝐴 = 0ℎ) |
4 | hsn0elch 31078 | . . . . . . . 8 ⊢ {0ℎ} ∈ Cℋ | |
5 | 4 | ococi 31235 | . . . . . . 7 ⊢ (⊥‘(⊥‘{0ℎ})) = {0ℎ} |
6 | 5 | eleq2i 2821 | . . . . . 6 ⊢ (𝐴 ∈ (⊥‘(⊥‘{0ℎ})) ↔ 𝐴 ∈ {0ℎ}) |
7 | h1de2.2 | . . . . . . . 8 ⊢ 𝐵 ∈ ℋ | |
8 | ax-hvmul0 30840 | . . . . . . . 8 ⊢ (𝐵 ∈ ℋ → (0 ·ℎ 𝐵) = 0ℎ) | |
9 | 7, 8 | ax-mp 5 | . . . . . . 7 ⊢ (0 ·ℎ 𝐵) = 0ℎ |
10 | 9 | eqeq2i 2741 | . . . . . 6 ⊢ (𝐴 = (0 ·ℎ 𝐵) ↔ 𝐴 = 0ℎ) |
11 | 3, 6, 10 | 3bitr4ri 303 | . . . . 5 ⊢ (𝐴 = (0 ·ℎ 𝐵) ↔ 𝐴 ∈ (⊥‘(⊥‘{0ℎ}))) |
12 | sneq 4642 | . . . . . . . 8 ⊢ (𝐵 = 0ℎ → {𝐵} = {0ℎ}) | |
13 | 12 | fveq2d 6906 | . . . . . . 7 ⊢ (𝐵 = 0ℎ → (⊥‘{𝐵}) = (⊥‘{0ℎ})) |
14 | 13 | fveq2d 6906 | . . . . . 6 ⊢ (𝐵 = 0ℎ → (⊥‘(⊥‘{𝐵})) = (⊥‘(⊥‘{0ℎ}))) |
15 | 14 | eleq2d 2815 | . . . . 5 ⊢ (𝐵 = 0ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 ∈ (⊥‘(⊥‘{0ℎ})))) |
16 | 11, 15 | bitr4id 289 | . . . 4 ⊢ (𝐵 = 0ℎ → (𝐴 = (0 ·ℎ 𝐵) ↔ 𝐴 ∈ (⊥‘(⊥‘{𝐵})))) |
17 | 0cn 11244 | . . . . 5 ⊢ 0 ∈ ℂ | |
18 | oveq1 7433 | . . . . . 6 ⊢ (𝑥 = 0 → (𝑥 ·ℎ 𝐵) = (0 ·ℎ 𝐵)) | |
19 | 18 | rspceeqv 3633 | . . . . 5 ⊢ ((0 ∈ ℂ ∧ 𝐴 = (0 ·ℎ 𝐵)) → ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵)) |
20 | 17, 19 | mpan 688 | . . . 4 ⊢ (𝐴 = (0 ·ℎ 𝐵) → ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵)) |
21 | 16, 20 | syl6bir 253 | . . 3 ⊢ (𝐵 = 0ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵))) |
22 | 1, 7 | h1de2bi 31384 | . . . 4 ⊢ (𝐵 ≠ 0ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵))) |
23 | his6 30929 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℋ → ((𝐵 ·ih 𝐵) = 0 ↔ 𝐵 = 0ℎ)) | |
24 | 7, 23 | ax-mp 5 | . . . . . . . 8 ⊢ ((𝐵 ·ih 𝐵) = 0 ↔ 𝐵 = 0ℎ) |
25 | 24 | necon3bii 2990 | . . . . . . 7 ⊢ ((𝐵 ·ih 𝐵) ≠ 0 ↔ 𝐵 ≠ 0ℎ) |
26 | 1, 7 | hicli 30911 | . . . . . . . 8 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ |
27 | 7, 7 | hicli 30911 | . . . . . . . 8 ⊢ (𝐵 ·ih 𝐵) ∈ ℂ |
28 | 26, 27 | divclzi 11987 | . . . . . . 7 ⊢ ((𝐵 ·ih 𝐵) ≠ 0 → ((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ∈ ℂ) |
29 | 25, 28 | sylbir 234 | . . . . . 6 ⊢ (𝐵 ≠ 0ℎ → ((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ∈ ℂ) |
30 | oveq1 7433 | . . . . . . 7 ⊢ (𝑥 = ((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) → (𝑥 ·ℎ 𝐵) = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)) | |
31 | 30 | rspceeqv 3633 | . . . . . 6 ⊢ ((((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ∈ ℂ ∧ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)) → ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵)) |
32 | 29, 31 | sylan 578 | . . . . 5 ⊢ ((𝐵 ≠ 0ℎ ∧ 𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵)) → ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵)) |
33 | 32 | ex 411 | . . . 4 ⊢ (𝐵 ≠ 0ℎ → (𝐴 = (((𝐴 ·ih 𝐵) / (𝐵 ·ih 𝐵)) ·ℎ 𝐵) → ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵))) |
34 | 22, 33 | sylbid 239 | . . 3 ⊢ (𝐵 ≠ 0ℎ → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵))) |
35 | 21, 34 | pm2.61ine 3022 | . 2 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) → ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵)) |
36 | snssi 4816 | . . . . . . . 8 ⊢ (𝐵 ∈ ℋ → {𝐵} ⊆ ℋ) | |
37 | occl 31134 | . . . . . . . 8 ⊢ ({𝐵} ⊆ ℋ → (⊥‘{𝐵}) ∈ Cℋ ) | |
38 | 7, 36, 37 | mp2b 10 | . . . . . . 7 ⊢ (⊥‘{𝐵}) ∈ Cℋ |
39 | 38 | choccli 31137 | . . . . . 6 ⊢ (⊥‘(⊥‘{𝐵})) ∈ Cℋ |
40 | 39 | chshii 31057 | . . . . 5 ⊢ (⊥‘(⊥‘{𝐵})) ∈ Sℋ |
41 | h1did 31381 | . . . . . 6 ⊢ (𝐵 ∈ ℋ → 𝐵 ∈ (⊥‘(⊥‘{𝐵}))) | |
42 | 7, 41 | ax-mp 5 | . . . . 5 ⊢ 𝐵 ∈ (⊥‘(⊥‘{𝐵})) |
43 | shmulcl 31048 | . . . . 5 ⊢ (((⊥‘(⊥‘{𝐵})) ∈ Sℋ ∧ 𝑥 ∈ ℂ ∧ 𝐵 ∈ (⊥‘(⊥‘{𝐵}))) → (𝑥 ·ℎ 𝐵) ∈ (⊥‘(⊥‘{𝐵}))) | |
44 | 40, 42, 43 | mp3an13 1448 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 ·ℎ 𝐵) ∈ (⊥‘(⊥‘{𝐵}))) |
45 | eleq1 2817 | . . . 4 ⊢ (𝐴 = (𝑥 ·ℎ 𝐵) → (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ (𝑥 ·ℎ 𝐵) ∈ (⊥‘(⊥‘{𝐵})))) | |
46 | 44, 45 | syl5ibrcom 246 | . . 3 ⊢ (𝑥 ∈ ℂ → (𝐴 = (𝑥 ·ℎ 𝐵) → 𝐴 ∈ (⊥‘(⊥‘{𝐵})))) |
47 | 46 | rexlimiv 3145 | . 2 ⊢ (∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵) → 𝐴 ∈ (⊥‘(⊥‘{𝐵}))) |
48 | 35, 47 | impbii 208 | 1 ⊢ (𝐴 ∈ (⊥‘(⊥‘{𝐵})) ↔ ∃𝑥 ∈ ℂ 𝐴 = (𝑥 ·ℎ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 ⊆ wss 3949 {csn 4632 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 0cc0 11146 / cdiv 11909 ℋchba 30749 ·ℎ csm 30751 ·ih csp 30752 0ℎc0v 30754 Sℋ csh 30758 Cℋ cch 30759 ⊥cort 30760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cc 10466 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226 ax-hilex 30829 ax-hfvadd 30830 ax-hvcom 30831 ax-hvass 30832 ax-hv0cl 30833 ax-hvaddid 30834 ax-hfvmul 30835 ax-hvmulid 30836 ax-hvmulass 30837 ax-hvdistr1 30838 ax-hvdistr2 30839 ax-hvmul0 30840 ax-hfi 30909 ax-his1 30912 ax-his2 30913 ax-his3 30914 ax-his4 30915 ax-hcompl 31032 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 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-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-oadd 8497 df-omul 8498 df-er 8731 df-map 8853 df-pm 8854 df-ixp 8923 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-fi 9442 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-acn 9973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 df-sum 15673 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-hom 17264 df-cco 17265 df-rest 17411 df-topn 17412 df-0g 17430 df-gsum 17431 df-topgen 17432 df-pt 17433 df-prds 17436 df-xrs 17491 df-qtop 17496 df-imas 17497 df-xps 17499 df-mre 17573 df-mrc 17574 df-acs 17576 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-submnd 18748 df-mulg 19031 df-cntz 19275 df-cmn 19744 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-cld 22943 df-ntr 22944 df-cls 22945 df-nei 23022 df-cn 23151 df-cnp 23152 df-lm 23153 df-haus 23239 df-tx 23486 df-hmeo 23679 df-fil 23770 df-fm 23862 df-flim 23863 df-flf 23864 df-xms 24246 df-ms 24247 df-tms 24248 df-cfil 25203 df-cau 25204 df-cmet 25205 df-grpo 30323 df-gid 30324 df-ginv 30325 df-gdiv 30326 df-ablo 30375 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-vs 30429 df-nmcv 30430 df-ims 30431 df-dip 30531 df-ssp 30552 df-ph 30643 df-cbn 30693 df-hnorm 30798 df-hba 30799 df-hvsub 30801 df-hlim 30802 df-hcau 30803 df-sh 31037 df-ch 31051 df-oc 31082 df-ch0 31083 |
This theorem is referenced by: h1de2ci 31386 |
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