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Theorem lmif1o 29115

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 29105	. . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
10	9	ffnd 6706	. 2 ⊢ (𝜑 → 𝑀 Fn 𝑃)
11	4	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐺 ∈ TarskiG)
12	5	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐺Dim_TarskiG≥2)
13	8	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝐷 ∈ ran 𝐿)
14		simpr 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → 𝑏 ∈ 𝑃)
15	1, 2, 3, 11, 12, 6, 7, 13, 14	lmilmi 29109	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → (𝑀‘(𝑀‘𝑏)) = 𝑏)
16	15	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏)
17		nvocnv 7279	. . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏) → ^◡𝑀 = 𝑀)
18	9, 16, 17	syl2anc 595	. 2 ⊢ (𝜑 → ^◡𝑀 = 𝑀)
19		nvof1o 7278	. 2 ⊢ ((𝑀 Fn 𝑃 ∧ ^◡𝑀 = 𝑀) → 𝑀:𝑃–1-1-onto→𝑃)
20	10, 18, 19	syl2anc 595	1 ⊢ (𝜑 → 𝑀:𝑃–1-1-onto→𝑃)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ∀wral 3078 class class class wbr 5108 ^◡ccnv 5659 ran crn 5661 Fn wfn 6531 ⟶wf 6532 –1-1-onto→wf1o 6535 ‘cfv 6536 2c2 12301 Basecbs 17275 distcds 17325 TarskiGcstrkg 28707 Dim_TarskiG≥cstrkgld 28711 Itvcitv 28713 LineGclng 28714 lInvGclmi 29093

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-oadd 8455 df-er 8692 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9894 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-3 12310 df-n0 12511 df-xnn0 12584 df-z 12598 df-uz 12869 df-fz 13542 df-fzo 13690 df-hash 14374 df-word 14558 df-concat 14615 df-s1 14641 df-s2 14892 df-s3 14893 df-trkgc 28728 df-trkgb 28729 df-trkgcb 28730 df-trkgld 28732 df-trkg 28733 df-cgrg 28791 df-leg 28863 df-mir 28941 df-rag 28985 df-perpg 28987 df-mid 29094 df-lmi 29095

This theorem is used by: lmimot 29118

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