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Theorem lmif1o 28729

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 28719	. . 3 ⊢ (𝜑 → 𝑀:𝑃⟶𝑃)
10	9	ffnd 6696	. 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 28723	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝑃) → (𝑀‘(𝑀‘𝑏)) = 𝑏)
16	15	ralrimiva 3127	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏)
17		nvocnv 7263	. . 3 ⊢ ((𝑀:𝑃⟶𝑃 ∧ ∀𝑏 ∈ 𝑃 (𝑀‘(𝑀‘𝑏)) = 𝑏) → ^◡𝑀 = 𝑀)
18	9, 16, 17	syl2anc 584	. 2 ⊢ (𝜑 → ^◡𝑀 = 𝑀)
19		nvof1o 7262	. 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 1540 ∈ wcel 2109 ∀wral 3046 class class class wbr 5115 ^◡ccnv 5645 ran crn 5647 Fn wfn 6514 ⟶wf 6515 –1-1-onto→wf1o 6518 ‘cfv 6519 2c2 12252 Basecbs 17185 distcds 17235 TarskiGcstrkg 28361 Dim_TarskiG≥cstrkgld 28365 Itvcitv 28367 LineGclng 28368 lInvGclmi 28707

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718 ax-cnex 11142 ax-resscn 11143 ax-1cn 11144 ax-icn 11145 ax-addcl 11146 ax-addrcl 11147 ax-mulcl 11148 ax-mulrcl 11149 ax-mulcom 11150 ax-addass 11151 ax-mulass 11152 ax-distr 11153 ax-i2m1 11154 ax-1ne0 11155 ax-1rid 11156 ax-rnegex 11157 ax-rrecex 11158 ax-cnre 11159 ax-pre-lttri 11160 ax-pre-lttrn 11161 ax-pre-ltadd 11162 ax-pre-mulgt0 11163

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-nel 3032 df-ral 3047 df-rex 3056 df-rmo 3357 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-pss 3942 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-tp 4602 df-op 4604 df-uni 4880 df-int 4919 df-iun 4965 df-br 5116 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5541 df-eprel 5546 df-po 5554 df-so 5555 df-fr 5599 df-we 5601 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-om 7851 df-1st 7977 df-2nd 7978 df-frecs 8269 df-wrecs 8300 df-recs 8349 df-rdg 8387 df-1o 8443 df-oadd 8447 df-er 8682 df-map 8805 df-pm 8806 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-dju 9872 df-card 9910 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-le 11232 df-sub 11425 df-neg 11426 df-nn 12198 df-2 12260 df-3 12261 df-n0 12459 df-xnn0 12532 df-z 12546 df-uz 12810 df-fz 13482 df-fzo 13629 df-hash 14306 df-word 14489 df-concat 14546 df-s1 14571 df-s2 14824 df-s3 14825 df-trkgc 28382 df-trkgb 28383 df-trkgcb 28384 df-trkgld 28386 df-trkg 28387 df-cgrg 28445 df-leg 28517 df-mir 28587 df-rag 28628 df-perpg 28630 df-mid 28708 df-lmi 28709

This theorem is referenced by: lmimot 28732

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