cdlemfnid - Mathbox for Norm Megill

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Theorem cdlemfnid 40563

Description: cdlemf 40562 with additional constraint of non-identity. (Contributed by NM, 20-Jun-2013.)

**Hypotheses**
Ref	Expression
cdlemfnid.b	⊢ 𝐵 = (Base‘𝐾)
cdlemfnid.l	⊢ ≤ = (le‘𝐾)
cdlemfnid.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemfnid.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemfnid.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemfnid.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)

**Assertion**
Ref	Expression
cdlemfnid	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵)))

Distinct variable groups: 𝐴,𝑓 𝑓,𝐻 𝑓,𝐾 ≤ ,𝑓 𝑇,𝑓 𝑈,𝑓 𝑓,𝑊 Allowed substitution hints: 𝐵(𝑓) 𝑅(𝑓)

**Proof of Theorem cdlemfnid**
Step	Hyp	Ref	Expression
1		cdlemfnid.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		cdlemfnid.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		cdlemfnid.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		cdlemfnid.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		cdlemfnid.r	. . 3 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	cdlemf 40562	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈)
7		simp3 1138	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → (𝑅‘𝑓) = 𝑈)
8		simp1rl 1239	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → 𝑈 ∈ 𝐴)
9	7, 8	eqeltrd 2828	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → (𝑅‘𝑓) ∈ 𝐴)
10		simp1l 1198	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		simp2 1137	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → 𝑓 ∈ 𝑇)
12		cdlemfnid.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
13	12, 2, 3, 4, 5	trlnidatb 40176	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝑓) ∈ 𝐴))
14	10, 11, 13	syl2anc 584	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝑓) ∈ 𝐴))
15	9, 14	mpbird 257	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → 𝑓 ≠ ( I ↾ 𝐵))
16	7, 15	jca 511	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵)))
17	16	3expia 1121	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((𝑅‘𝑓) = 𝑈 → ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵))))
18	17	reximdva 3142	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → (∃𝑓 ∈ 𝑇 (𝑅‘𝑓) = 𝑈 → ∃𝑓 ∈ 𝑇 ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵))))
19	6, 18	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 ((𝑅‘𝑓) = 𝑈 ∧ 𝑓 ≠ ( I ↾ 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∃wrex 3053 class class class wbr 5095 I cid 5517 ↾ cres 5625 ‘cfv 6486 Basecbs 17139 lecple 17187 Atomscatm 39261 HLchlt 39348 LHypclh 39983 LTrncltrn 40100 trLctrl 40157

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 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-riotaBAD 38951

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7931 df-2nd 7932 df-undef 8213 df-map 8762 df-proset 18219 df-poset 18238 df-plt 18253 df-lub 18269 df-glb 18270 df-join 18271 df-meet 18272 df-p0 18348 df-p1 18349 df-lat 18357 df-clat 18424 df-oposet 39174 df-ol 39176 df-oml 39177 df-covers 39264 df-ats 39265 df-atl 39296 df-cvlat 39320 df-hlat 39349 df-llines 39497 df-lplanes 39498 df-lvols 39499 df-lines 39500 df-psubsp 39502 df-pmap 39503 df-padd 39795 df-lhyp 39987 df-laut 39988 df-ldil 40103 df-ltrn 40104 df-trl 40158

This theorem is referenced by: cdlemftr3 40564 cdlemj3 40822

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