cdlemfnid - Mathbox for Norm Megill

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Theorem cdlemfnid 40521

Description: cdlemf 40520 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 40520	. 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 1238	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → 𝑈 ∈ 𝐴)
9	7, 8	eqeltrd 2844	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) = 𝑈) → (𝑅‘𝑓) ∈ 𝐴)
10		simp1l 1197	. . . . . . 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 40134	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (𝑅‘𝑓) ∈ 𝐴))
14	10, 11, 13	syl2anc 583	. . . . . 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 3174	. 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 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∃wrex 3076 class class class wbr 5166 I cid 5592 ↾ cres 5702 ‘cfv 6573 Basecbs 17258 lecple 17318 Atomscatm 39219 HLchlt 39306 LHypclh 39941 LTrncltrn 40058 trLctrl 40115

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-riotaBAD 38909

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-undef 8314 df-map 8886 df-proset 18365 df-poset 18383 df-plt 18400 df-lub 18416 df-glb 18417 df-join 18418 df-meet 18419 df-p0 18495 df-p1 18496 df-lat 18502 df-clat 18569 df-oposet 39132 df-ol 39134 df-oml 39135 df-covers 39222 df-ats 39223 df-atl 39254 df-cvlat 39278 df-hlat 39307 df-llines 39455 df-lplanes 39456 df-lvols 39457 df-lines 39458 df-psubsp 39460 df-pmap 39461 df-padd 39753 df-lhyp 39945 df-laut 39946 df-ldil 40061 df-ltrn 40062 df-trl 40116

This theorem is referenced by: cdlemftr3 40522 cdlemj3 40780

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