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| Mirrors > Home > MPE Home > Th. List > lmicom | Structured version Visualization version GIF version | ||
| Description: The line mirroring function is an involution. Theorem 10.4 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.) |
| Ref | Expression |
|---|---|
| ismid.p | ⊢ 𝑃 = (Base‘𝐺) |
| ismid.d | ⊢ − = (dist‘𝐺) |
| ismid.i | ⊢ 𝐼 = (Itv‘𝐺) |
| ismid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| ismid.1 | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
| lmif.m | ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) |
| lmif.l | ⊢ 𝐿 = (LineG‘𝐺) |
| lmif.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
| lmicl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
| islmib.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
| lmicom.1 | ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵) |
| Ref | Expression |
|---|---|
| lmicom | ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ismid.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
| 2 | ismid.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
| 3 | ismid.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
| 4 | ismid.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 5 | ismid.1 | . . . . 5 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
| 6 | lmicl.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 7 | islmib.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | midcom 28790 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = (𝐵(midG‘𝐺)𝐴)) |
| 9 | lmicom.1 | . . . . . . 7 ⊢ (𝜑 → (𝑀‘𝐴) = 𝐵) | |
| 10 | 9 | eqcomd 2743 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (𝑀‘𝐴)) |
| 11 | lmif.m | . . . . . . 7 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) | |
| 12 | lmif.l | . . . . . . 7 ⊢ 𝐿 = (LineG‘𝐺) | |
| 13 | lmif.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
| 14 | 1, 2, 3, 4, 5, 11, 12, 13, 6, 7 | islmib 28795 | . . . . . 6 ⊢ (𝜑 → (𝐵 = (𝑀‘𝐴) ↔ ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)))) |
| 15 | 10, 14 | mpbid 232 | . . . . 5 ⊢ (𝜑 → ((𝐴(midG‘𝐺)𝐵) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵))) |
| 16 | 15 | simpld 494 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝐷) |
| 17 | 8, 16 | eqeltrrd 2842 | . . 3 ⊢ (𝜑 → (𝐵(midG‘𝐺)𝐴) ∈ 𝐷) |
| 18 | 15 | simprd 495 | . . . . . . . . 9 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ∨ 𝐴 = 𝐵)) |
| 19 | 18 | orcomd 872 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵))) |
| 20 | 19 | ord 865 | . . . . . . 7 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵))) |
| 21 | 4 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG) |
| 22 | 6 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ∈ 𝑃) |
| 23 | 7 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃) |
| 24 | simpr 484 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → ¬ 𝐴 = 𝐵) | |
| 25 | 24 | neqned 2947 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → 𝐴 ≠ 𝐵) |
| 26 | 1, 3, 12, 21, 22, 23, 25 | tglinecom 28643 | . . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐴𝐿𝐵) = (𝐵𝐿𝐴)) |
| 27 | 26 | breq2d 5155 | . . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐴 = 𝐵) → (𝐷(⟂G‘𝐺)(𝐴𝐿𝐵) ↔ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
| 28 | 27 | pm5.74da 804 | . . . . . . 7 ⊢ (𝜑 → ((¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐴𝐿𝐵)) ↔ (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴)))) |
| 29 | 20, 28 | mpbid 232 | . . . . . 6 ⊢ (𝜑 → (¬ 𝐴 = 𝐵 → 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
| 30 | 29 | orrd 864 | . . . . 5 ⊢ (𝜑 → (𝐴 = 𝐵 ∨ 𝐷(⟂G‘𝐺)(𝐵𝐿𝐴))) |
| 31 | 30 | orcomd 872 | . . . 4 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵)) |
| 32 | eqcom 2744 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ 𝐵 = 𝐴) | |
| 33 | 32 | orbi2i 913 | . . . 4 ⊢ ((𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐴 = 𝐵) ↔ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)) |
| 34 | 31, 33 | sylib 218 | . . 3 ⊢ (𝜑 → (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)) |
| 35 | 1, 2, 3, 4, 5, 11, 12, 13, 7, 6 | islmib 28795 | . . 3 ⊢ (𝜑 → (𝐴 = (𝑀‘𝐵) ↔ ((𝐵(midG‘𝐺)𝐴) ∈ 𝐷 ∧ (𝐷(⟂G‘𝐺)(𝐵𝐿𝐴) ∨ 𝐵 = 𝐴)))) |
| 36 | 17, 34, 35 | mpbir2and 713 | . 2 ⊢ (𝜑 → 𝐴 = (𝑀‘𝐵)) |
| 37 | 36 | eqcomd 2743 | 1 ⊢ (𝜑 → (𝑀‘𝐵) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ran crn 5686 ‘cfv 6561 (class class class)co 7431 2c2 12321 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 DimTarskiG≥cstrkgld 28439 Itvcitv 28441 LineGclng 28442 ⟂Gcperpg 28703 midGcmid 28780 lInvGclmi 28781 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-dju 9941 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-xnn0 12600 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-s2 14887 df-s3 14888 df-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkgld 28460 df-trkg 28461 df-cgrg 28519 df-leg 28591 df-mir 28661 df-rag 28702 df-perpg 28704 df-mid 28782 df-lmi 28783 |
| This theorem is referenced by: lmilmi 28797 |
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