prjspnn0 - Mathbox for Steven Nguyen

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Theorem prjspnn0 42861

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 2736	. . . 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 2736	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2736	. . . 4 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
6		prjspnssbas.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ DivRing)
7	1, 2, 3, 4, 5, 6	prjspner 42858	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))} Er 𝐵)
8		erdm 8645	. . 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 42857	. . . . 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 2783	. . 3 ⊢ (𝜑 → 𝑃 = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
16	10, 15	eleqtrd 2838	. 2 ⊢ (𝜑 → 𝐴 ∈ (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘𝑊)𝑦))}))
17		elqsn0 8721	. 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 1541 ∈ wcel 2113 ≠ wne 2932 ∃wrex 3060 ∖ cdif 3898 ∅c0 4285 {csn 4580 {copab 5160 dom cdm 5624 ‘cfv 6492 (class class class)co 7358 Er wer 8632 / cqs 8634 0cc0 11026 ℕ₀cn0 12401 ...cfz 13423 Basecbs 17136 ·_𝑠 cvsca 17181 0_gc0g 17359 DivRingcdr 20662 freeLMod cfrlm 21701 ℙ𝕣𝕠𝕛_ncprjspn 42853

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7809 df-1st 7933 df-2nd 7934 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-ec 8637 df-qs 8641 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-hom 17201 df-cco 17202 df-0g 17361 df-prds 17367 df-pws 17369 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-subrg 20503 df-drng 20664 df-lmod 20813 df-lss 20883 df-lvec 21055 df-sra 21125 df-rgmod 21126 df-dsmm 21687 df-frlm 21702 df-prjsp 42841 df-prjspn 42854

This theorem is referenced by: prjcrv0 42872

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