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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 26557 | . . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃) |
10 | 9 | ffnd 6501 | . 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 26561 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → (𝑀‘(𝑀‘𝑏)) = 𝑏) |
16 | 15 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏) |
17 | nvocnv 7024 | . . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏) → ◡𝑀 = 𝑀) | |
18 | 9, 16, 17 | syl2anc 586 | . 2 ⊢ (𝜑 → ◡𝑀 = 𝑀) |
19 | nvof1o 7023 | . 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 1537 ∈ wcel 2114 ∀wral 3138 class class class wbr 5052 ◡ccnv 5540 ran crn 5542 Fn wfn 6336 ⟶wf 6337 –1-1-onto→wf1o 6340 ‘cfv 6341 2c2 11679 Basecbs 16466 distcds 16557 TarskiGcstrkg 26202 DimTarskiG≥cstrkgld 26206 Itvcitv 26208 LineGclng 26209 lInvGclmi 26545 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-dju 9316 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-n0 11885 df-xnn0 11955 df-z 11969 df-uz 12231 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-concat 13908 df-s1 13935 df-s2 14195 df-s3 14196 df-trkgc 26220 df-trkgb 26221 df-trkgcb 26222 df-trkgld 26224 df-trkg 26225 df-cgrg 26283 df-leg 26355 df-mir 26425 df-rag 26466 df-perpg 26468 df-mid 26546 df-lmi 26547 |
This theorem is referenced by: lmimot 26570 |
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