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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 27725 | . . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
10 | 9 | ffnd 6669 | . 2 ⊢ (𝜑 → 𝑀 Fn 𝑃) |
11 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐺 ∈ TarskiG) |
12 | 5 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐺DimTarskiG≥2) |
13 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐷 ∈ ran 𝐿) |
14 | simpr 485 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝑏 ∈ 𝑃) | |
15 | 1, 2, 3, 11, 12, 6, 7, 13, 14 | lmilmi 27729 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → (𝑀‘(𝑀‘𝑏)) = 𝑏) |
16 | 15 | ralrimiva 3143 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏) |
17 | nvocnv 7226 | . . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏) → ◡𝑀 = 𝑀) | |
18 | 9, 16, 17 | syl2anc 584 | . 2 ⊢ (𝜑 → ◡𝑀 = 𝑀) |
19 | nvof1o 7225 | . 2 ⊢ ((𝑀 Fn 𝑃 ∧ ◡𝑀 = 𝑀) → 𝑀:𝑃–1-1-onto→𝑃) | |
20 | 10, 18, 19 | syl2anc 584 | 1 ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 class class class wbr 5105 ◡ccnv 5632 ran crn 5634 Fn wfn 6491 ⟶wf 6492 –1-1-onto→wf1o 6495 ‘cfv 6496 2c2 12207 Basecbs 17082 distcds 17141 TarskiGcstrkg 27367 DimTarskiG≥cstrkgld 27371 Itvcitv 27373 LineGclng 27374 lInvGclmi 27713 |
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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-oadd 8415 df-er 8647 df-map 8766 df-pm 8767 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-dju 9836 df-card 9874 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-n0 12413 df-xnn0 12485 df-z 12499 df-uz 12763 df-fz 13424 df-fzo 13567 df-hash 14230 df-word 14402 df-concat 14458 df-s1 14483 df-s2 14736 df-s3 14737 df-trkgc 27388 df-trkgb 27389 df-trkgcb 27390 df-trkgld 27392 df-trkg 27393 df-cgrg 27451 df-leg 27523 df-mir 27593 df-rag 27634 df-perpg 27636 df-mid 27714 df-lmi 27715 |
This theorem is referenced by: lmimot 27738 |
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