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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sseqmw | Structured version Visualization version GIF version |
Description: Lemma for sseqf 33379 amd sseqp1 33382. (Contributed by Thierry Arnoux, 25-Apr-2019.) |
Ref | Expression |
---|---|
sseqval.1 | ā¢ (š ā š ā V) |
sseqval.2 | ā¢ (š ā š ā Word š) |
sseqval.3 | ā¢ š = (Word š ā© (ā”āÆ ā (ā¤ā„ā(āÆāš)))) |
sseqval.4 | ā¢ (š ā š¹:šā¶š) |
Ref | Expression |
---|---|
sseqmw | ā¢ (š ā š ā š) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseqval.2 | . . 3 ā¢ (š ā š ā Word š) | |
2 | elex 3492 | . . . . 5 ā¢ (š ā Word š ā š ā V) | |
3 | 1, 2 | syl 17 | . . . 4 ā¢ (š ā š ā V) |
4 | lencl 14479 | . . . . . 6 ā¢ (š ā Word š ā (āÆāš) ā ā0) | |
5 | 4 | nn0zd 12580 | . . . . 5 ā¢ (š ā Word š ā (āÆāš) ā ā¤) |
6 | uzid 12833 | . . . . 5 ā¢ ((āÆāš) ā ā¤ ā (āÆāš) ā (ā¤ā„ā(āÆāš))) | |
7 | 1, 5, 6 | 3syl 18 | . . . 4 ā¢ (š ā (āÆāš) ā (ā¤ā„ā(āÆāš))) |
8 | hashf 14294 | . . . . 5 ā¢ āÆ:Vā¶(ā0 āŖ {+ā}) | |
9 | ffn 6714 | . . . . 5 ā¢ (āÆ:Vā¶(ā0 āŖ {+ā}) ā āÆ Fn V) | |
10 | elpreima 7056 | . . . . 5 ā¢ (āÆ Fn V ā (š ā (ā”āÆ ā (ā¤ā„ā(āÆāš))) ā (š ā V ā§ (āÆāš) ā (ā¤ā„ā(āÆāš))))) | |
11 | 8, 9, 10 | mp2b 10 | . . . 4 ā¢ (š ā (ā”āÆ ā (ā¤ā„ā(āÆāš))) ā (š ā V ā§ (āÆāš) ā (ā¤ā„ā(āÆāš)))) |
12 | 3, 7, 11 | sylanbrc 583 | . . 3 ā¢ (š ā š ā (ā”āÆ ā (ā¤ā„ā(āÆāš)))) |
13 | 1, 12 | elind 4193 | . 2 ā¢ (š ā š ā (Word š ā© (ā”āÆ ā (ā¤ā„ā(āÆāš))))) |
14 | sseqval.3 | . 2 ā¢ š = (Word š ā© (ā”āÆ ā (ā¤ā„ā(āÆāš)))) | |
15 | 13, 14 | eleqtrrdi 2844 | 1 ā¢ (š ā š ā š) |
Colors of variables: wff setvar class |
Syntax hints: ā wi 4 ā wb 205 ā§ wa 396 = wceq 1541 ā wcel 2106 Vcvv 3474 āŖ cun 3945 ā© cin 3946 {csn 4627 ā”ccnv 5674 ā cima 5678 Fn wfn 6535 ā¶wf 6536 ācfv 6540 +ācpnf 11241 ā0cn0 12468 ā¤cz 12554 ā¤ā„cuz 12818 āÆchash 14286 Word cword 14460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-xnn0 12541 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 |
This theorem is referenced by: sseqf 33379 |
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