prjspnn0 - Mathbox for Steven Nguyen

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Theorem prjspnn0 43631

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

**Hypotheses**
Ref	Expression
prjspnn0.p	⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛_n𝐾)
prjspnn0.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
prjspnn0.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 2761	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))}
2		eqid 2761	. . . 4 ⊢ (𝐾 freeLMod (0...𝑁)) = (𝐾 freeLMod (0...𝑁))
3		eqid 2761	. . . 4 ⊢ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) = ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})
4		eqid 2761	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
5		eqid 2761	. . . 4 ⊢ ( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁))) = ( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))
6		prjspnn0.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ DivRing)
7	1, 2, 3, 4, 5, 6	prjspner 43627	. . 3 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))} Er ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}))
8		erdm 8721	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))} Er ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))} = ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}))
9	7, 8	syl 18	. 2 ⊢ (𝜑 → dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))} = ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}))
10		prjspnn0.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
11		prjspnn0.p	. . . 4 ⊢ 𝑃 = (𝑁ℙ𝕣𝕠𝕛_n𝐾)
12		prjspnn0.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
13	1, 2, 3, 4, 5, 12, 6	prjspnval2 43626	. . . 4 ⊢ (𝜑 → (𝑁ℙ𝕣𝕠𝕛_n𝐾) = (((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))}))
14	11, 13	eqtrid 2808	. . 3 ⊢ (𝜑 → 𝑃 = (((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))}))
15	10, 14	eleqtrd 2863	. 2 ⊢ (𝜑 → 𝐴 ∈ (((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))}))
16		elqsn0 8798	. 2 ⊢ ((dom {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))} = ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝐴 ∈ (((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) / {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))}) ∧ 𝑦 ∈ ((Base‘(𝐾 freeLMod (0...𝑁))) ∖ {(0_g‘(𝐾 freeLMod (0...𝑁)))})) ∧ ∃𝑙 ∈ (Base‘𝐾)𝑥 = (𝑙( ·_𝑠 ‘(𝐾 freeLMod (0...𝑁)))𝑦))})) → 𝐴 ≠ ∅)
17	9, 15, 16	syl2anc 596	1 ⊢ (𝜑 → 𝐴 ≠ ∅)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∃wrex 3087 ∖ cdif 3896 ∅c0 4279 {csn 4584 {copab 5167 dom cdm 5651 ‘cfv 6537 (class class class)co 7418 Er wer 8707 / cqs 8709 0cc0 11193 ℕ₀cn0 12599 ...cfz 13632 Basecbs 17380 ·_𝑠 cvsca 17425 0_gc0g 17603 DivRingcdr 20973 freeLMod cfrlm 22045 ℙ𝕣𝕠𝕛_ncprjspn 43622

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-tpos 8236 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-ec 8712 df-qs 8716 df-map 8842 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-sup 9427 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-fz 13633 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-hom 17445 df-cco 17446 df-0g 17605 df-prds 17611 df-pws 17613 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-grp 19140 df-minusg 19141 df-sbg 19142 df-subg 19326 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-subrg 20815 df-drng 20975 df-lmod 21130 df-lss 21200 df-lvec 21371 df-sra 21441 df-rgmod 21442 df-dsmm 22031 df-frlm 22046 df-prjsp 43610 df-prjspn 43623

This theorem is used by: prjcrv0 43649

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