prjspnn0 - Mathbox for Steven Nguyen

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Theorem prjspnn0 42617

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 42614	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} Er 𝐵)
8		erdm 8684	. . 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 42613	. . . . 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 584	. . . 4 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
15	11, 14	eqtrid 2777	. . 3 ⊢ (𝜑 → 𝑃 = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
16	10, 15	eleqtrd 2831	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
17		elqsn0 8760	. 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 584	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 ∃wrex 3054 ∖ cdif 3914 ∅c0 4299 {csn 4592 {copab 5172 dom cdm 5641 ‘cfv 6514 (class class class)co 7390 Er wer 8671 / cqs 8673 0cc0 11075 ℕ₀cn0 12449 ...cfz 13475 Basecbs 17186 ·_𝑠 cvsca 17231 0_gc0g 17409 DivRingcdr 20645 freeLMod cfrlm 21662 ℙ𝕣𝕠𝕛_ncprjspn 42609

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 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-ec 8676 df-qs 8680 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-sup 9400 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17411 df-prds 17417 df-pws 17419 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-subrg 20486 df-drng 20647 df-lmod 20775 df-lss 20845 df-lvec 21017 df-sra 21087 df-rgmod 21088 df-dsmm 21648 df-frlm 21663 df-prjsp 42597 df-prjspn 42610

This theorem is referenced by: prjcrv0 42628

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