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Theorem lmif1o 28829

Description: The line mirroring function 𝑀 is a bijection. Theorem 10.9 of [Schwabhauser] p. 89. (Contributed by Thierry Arnoux, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
ismid.p	⊢ 𝑃 = (Base‘𝐺)
ismid.d	⊢ − = (dist‘𝐺)
ismid.i	⊢ 𝐼 = (Itv‘𝐺)
ismid.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
ismid.1	⊢ (𝜑 → 𝐺Dim_TarskiG≥2)
lmif.m	⊢ 𝑀 = ((lInvG‘𝐺)‘𝐷)
lmif.l	⊢ 𝐿 = (LineG‘𝐺)
lmif.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)

**Assertion**
Ref	Expression
lmif1o	⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃)

Proof of Theorem lmif1o Dummy variable 𝑏 is distinct from all other variables.

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 → 𝐺Dim_TarskiG≥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 28819	. . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
10	9	ffnd 6745	. 2 ⊢ (𝜑 → 𝑀 Fn 𝑃)
11	4	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐺 ∈ TarskiG)
12	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐺Dim_TarskiG≥2)
13	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐷 ∈ ran 𝐿)
14		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝑏 ∈ 𝑃)
15	1, 2, 3, 11, 12, 6, 7, 13, 14	lmilmi 28823	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → (𝑀‘(𝑀‘𝑏)) = 𝑏)
16	15	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏)
17		nvocnv 7308	. . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏) → ^◡𝑀 = 𝑀)
18	9, 16, 17	syl2anc 584	. 2 ⊢ (𝜑 → ^◡𝑀 = 𝑀)
19		nvof1o 7307	. 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 395 = wceq 1539 ∈ wcel 2108 ∀wral 3061 class class class wbr 5151 ^◡ccnv 5692 ran crn 5694 Fn wfn 6564 ⟶wf 6565 –1-1-onto→wf1o 6568 ‘cfv 6569 2c2 12328 Basecbs 17254 distcds 17316 TarskiGcstrkg 28461 Dim_TarskiG≥cstrkgld 28465 Itvcitv 28467 LineGclng 28468 lInvGclmi 28807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 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 5288 ax-sep 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 ax-un 7761 ax-cnex 11218 ax-resscn 11219 ax-1cn 11220 ax-icn 11221 ax-addcl 11222 ax-addrcl 11223 ax-mulcl 11224 ax-mulrcl 11225 ax-mulcom 11226 ax-addass 11227 ax-mulass 11228 ax-distr 11229 ax-i2m1 11230 ax-1ne0 11231 ax-1rid 11232 ax-rnegex 11233 ax-rrecex 11234 ax-cnre 11235 ax-pre-lttri 11236 ax-pre-lttrn 11237 ax-pre-ltadd 11238 ax-pre-mulgt0 11239

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 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 3483 df-sbc 3795 df-csb 3912 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-pss 3986 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-tp 4639 df-op 4641 df-uni 4916 df-int 4955 df-iun 5001 df-br 5152 df-opab 5214 df-mpt 5235 df-tr 5269 df-id 5587 df-eprel 5593 df-po 5601 df-so 5602 df-fr 5645 df-we 5647 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-pred 6329 df-ord 6395 df-on 6396 df-lim 6397 df-suc 6398 df-iota 6522 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-fv 6577 df-riota 7395 df-ov 7441 df-oprab 7442 df-mpo 7443 df-om 7895 df-1st 8022 df-2nd 8023 df-frecs 8314 df-wrecs 8345 df-recs 8419 df-rdg 8458 df-1o 8514 df-oadd 8518 df-er 8753 df-map 8876 df-pm 8877 df-en 8994 df-dom 8995 df-sdom 8996 df-fin 8997 df-dju 9948 df-card 9986 df-pnf 11304 df-mnf 11305 df-xr 11306 df-ltxr 11307 df-le 11308 df-sub 11501 df-neg 11502 df-nn 12274 df-2 12336 df-3 12337 df-n0 12534 df-xnn0 12607 df-z 12621 df-uz 12886 df-fz 13554 df-fzo 13701 df-hash 14376 df-word 14559 df-concat 14615 df-s1 14640 df-s2 14893 df-s3 14894 df-trkgc 28482 df-trkgb 28483 df-trkgcb 28484 df-trkgld 28486 df-trkg 28487 df-cgrg 28545 df-leg 28617 df-mir 28687 df-rag 28728 df-perpg 28730 df-mid 28808 df-lmi 28809

This theorem is referenced by: lmimot 28832

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