prjspnn0 - Mathbox for Steven Nguyen

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Theorem prjspnn0 41666

Description: A projective point is nonempty. (Contributed by SN, 17-Jan-2025.)

**Hypotheses**
Ref	Expression
prjspnssbas.p	⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛_n𝐾)
prjspnssbas.w	⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
prjspnssbas.b	⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
prjspnssbas.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
prjspnssbas.k	⊢ (𝜑 → 𝐾 ∈ DivRing)
prjspnn0.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)

**Assertion**
Ref	Expression
prjspnn0	⊢ (𝜑 → 𝐴 ≠ ∅)

Proof of Theorem prjspnn0 Dummy variables 𝑙 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2730	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}
2		prjspnssbas.w	. . . 4 ⊢ 𝑊 = (𝐾 freeLMod (0...𝑁))
3		prjspnssbas.b	. . . 4 ⊢ 𝐵 = ((Base‘𝑊) ∖ {(0_g‘𝑊)})
4		eqid 2730	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2730	. . . 4 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
6		prjspnssbas.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ DivRing)
7	1, 2, 3, 4, 5, 6	prjspner 41663	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} Er 𝐵)
8		erdm 8715	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} Er 𝐵 → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} = 𝐵)
9	7, 8	syl 17	. 2 ⊢ (𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} = 𝐵)
10		prjspnn0.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11		prjspnssbas.p	. . . 4 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛_n𝐾)
12		prjspnssbas.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
13	1, 2, 3, 4, 5	prjspnval2 41662	. . . . 5 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝐾 ∈ DivRing) → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
14	12, 6, 13	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
15	11, 14	eqtrid 2782	. . 3 ⊢ (𝜑 → 𝑃 = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
16	10, 15	eleqtrd 2833	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
17		elqsn0 8782	. 2 ⊢ ((dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} = 𝐵 ∧ 𝐴 ∈ (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))})) → 𝐴 ≠ ∅)
18	9, 16, 17	syl2anc 582	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∃wrex 3068 ∖ cdif 3944 ∅c0 4321 {csn 4627 {copab 5209 dom cdm 5675 ‘cfv 6542 (class class class)co 7411 Er wer 8702 / cqs 8704 0cc0 11112 ℕ₀cn0 12476 ...cfz 13488 Basecbs 17148 ·_𝑠 cvsca 17205 0_gc0g 17389 DivRingcdr 20500 freeLMod cfrlm 21520 ℙ𝕣𝕠𝕛_ncprjspn 41658

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13489 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-sca 17217 df-vsca 17218 df-ip 17219 df-tset 17220 df-ple 17221 df-ds 17223 df-hom 17225 df-cco 17226 df-0g 17391 df-prds 17397 df-pws 17399 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19039 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-subrg 20459 df-drng 20502 df-lmod 20616 df-lss 20687 df-lvec 20858 df-sra 20930 df-rgmod 20931 df-dsmm 21506 df-frlm 21521 df-prjsp 41646 df-prjspn 41659

This theorem is referenced by: prjcrv0 41677

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