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Mirrors > Home > MPE Home > Th. List > lmif1o | Structured version Visualization version GIF version |
Description: The line mirroring function 𝑀 is a bijection. Theorem 10.9 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) |
Ref | Expression |
---|---|
lmif1o | ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismid.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | ismid.d | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | ismid.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | ismid.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | ismid.1 | . . . 4 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
6 | lmif.m | . . . 4 ⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷) | |
7 | lmif.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
8 | lmif.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | lmif 26565 | . . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
10 | 9 | ffnd 6510 | . 2 ⊢ (𝜑 → 𝑀 Fn 𝑃) |
11 | 4 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐺 ∈ TarskiG) |
12 | 5 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐺DimTarskiG≥2) |
13 | 8 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐷 ∈ ran 𝐿) |
14 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝑏 ∈ 𝑃) | |
15 | 1, 2, 3, 11, 12, 6, 7, 13, 14 | lmilmi 26569 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → (𝑀‘(𝑀‘𝑏)) = 𝑏) |
16 | 15 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏) |
17 | nvocnv 7032 | . . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏) → ◡𝑀 = 𝑀) | |
18 | 9, 16, 17 | syl2anc 586 | . 2 ⊢ (𝜑 → ◡𝑀 = 𝑀) |
19 | nvof1o 7031 | . 2 ⊢ ((𝑀 Fn 𝑃 ∧ ◡𝑀 = 𝑀) → 𝑀:𝑃–1-1-onto→𝑃) | |
20 | 10, 18, 19 | syl2anc 586 | 1 ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5059 ◡ccnv 5549 ran crn 5551 Fn wfn 6345 ⟶wf 6346 –1-1-onto→wf1o 6349 ‘cfv 6350 2c2 11686 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 DimTarskiG≥cstrkgld 26214 Itvcitv 26216 LineGclng 26217 lInvGclmi 26553 |
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-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 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 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-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-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-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-dju 9324 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-s2 14204 df-s3 14205 df-trkgc 26228 df-trkgb 26229 df-trkgcb 26230 df-trkgld 26232 df-trkg 26233 df-cgrg 26291 df-leg 26363 df-mir 26433 df-rag 26474 df-perpg 26476 df-mid 26554 df-lmi 26555 |
This theorem is referenced by: lmimot 26578 |
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