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Mirrors > Home > MPE Home > Th. List > lmieq | Structured version Visualization version GIF version |
Description: Equality deduction for line mirroring. Theorem 10.7 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
Ref | Expression |
---|---|
ismid.p | β’ π = (BaseβπΊ) |
ismid.d | β’ β = (distβπΊ) |
ismid.i | β’ πΌ = (ItvβπΊ) |
ismid.g | β’ (π β πΊ β TarskiG) |
ismid.1 | β’ (π β πΊDimTarskiGβ₯2) |
lmif.m | β’ π = ((lInvGβπΊ)βπ·) |
lmif.l | β’ πΏ = (LineGβπΊ) |
lmif.d | β’ (π β π· β ran πΏ) |
lmicl.1 | β’ (π β π΄ β π) |
lmieq.c | β’ (π β π΅ β π) |
lmieq.d | β’ (π β (πβπ΄) = (πβπ΅)) |
Ref | Expression |
---|---|
lmieq | β’ (π β π΄ = π΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6900 | . 2 β’ (π = π΄ β ((πβπ) = (πβπ΅) β (πβπ΄) = (πβπ΅))) | |
2 | fveqeq2 6900 | . 2 β’ (π = π΅ β ((πβπ) = (πβπ΅) β (πβπ΅) = (πβπ΅))) | |
3 | ismid.p | . . 3 β’ π = (BaseβπΊ) | |
4 | ismid.d | . . 3 β’ β = (distβπΊ) | |
5 | ismid.i | . . 3 β’ πΌ = (ItvβπΊ) | |
6 | ismid.g | . . 3 β’ (π β πΊ β TarskiG) | |
7 | ismid.1 | . . 3 β’ (π β πΊDimTarskiGβ₯2) | |
8 | lmif.m | . . 3 β’ π = ((lInvGβπΊ)βπ·) | |
9 | lmif.l | . . 3 β’ πΏ = (LineGβπΊ) | |
10 | lmif.d | . . 3 β’ (π β π· β ran πΏ) | |
11 | lmieq.c | . . . 4 β’ (π β π΅ β π) | |
12 | 3, 4, 5, 6, 7, 8, 9, 10, 11 | lmicl 28470 | . . 3 β’ (π β (πβπ΅) β π) |
13 | 3, 4, 5, 6, 7, 8, 9, 10, 12 | lmireu 28474 | . 2 β’ (π β β!π β π (πβπ) = (πβπ΅)) |
14 | lmicl.1 | . 2 β’ (π β π΄ β π) | |
15 | lmieq.d | . 2 β’ (π β (πβπ΄) = (πβπ΅)) | |
16 | eqidd 2732 | . 2 β’ (π β (πβπ΅) = (πβπ΅)) | |
17 | 1, 2, 13, 14, 11, 15, 16 | reu2eqd 3732 | 1 β’ (π β π΄ = π΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 class class class wbr 5148 ran crn 5677 βcfv 6543 2c2 12274 Basecbs 17151 distcds 17213 TarskiGcstrkg 28111 DimTarskiGβ₯cstrkgld 28115 Itvcitv 28117 LineGclng 28118 lInvGclmi 28457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-oadd 8476 df-er 8709 df-map 8828 df-pm 8829 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-dju 9902 df-card 9940 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-n0 12480 df-xnn0 12552 df-z 12566 df-uz 12830 df-fz 13492 df-fzo 13635 df-hash 14298 df-word 14472 df-concat 14528 df-s1 14553 df-s2 14806 df-s3 14807 df-trkgc 28132 df-trkgb 28133 df-trkgcb 28134 df-trkgld 28136 df-trkg 28137 df-cgrg 28195 df-leg 28267 df-mir 28337 df-rag 28378 df-perpg 28380 df-mid 28458 df-lmi 28459 |
This theorem is referenced by: trgcopyeulem 28489 |
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