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Theorem pzriprnglem7 21406

Description: Lemma 7 for pzriprng 21416: 𝐽 is a unital ring. (Contributed by AV, 19-Mar-2025.)

**Hypotheses**
Ref	Expression
pzriprng.r	⊢ 𝑅 = (ℤ_ring ×_s ℤ_ring)
pzriprng.i	⊢ 𝐼 = (ℤ × {0})
pzriprng.j	⊢ 𝐽 = (𝑅 ↾_s 𝐼)

**Assertion**
Ref	Expression
pzriprnglem7	⊢ 𝐽 ∈ Ring

Proof of Theorem pzriprnglem7 Dummy variables 𝑥 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pzriprng.r	. . . 4 ⊢ 𝑅 = (ℤ_ring ×_s ℤ_ring)
2		pzriprng.i	. . . 4 ⊢ 𝐼 = (ℤ × {0})
3	1, 2	pzriprnglem5 21404	. . 3 ⊢ 𝐼 ∈ (SubRng‘𝑅)
4		pzriprng.j	. . . 4 ⊢ 𝐽 = (𝑅 ↾_s 𝐼)
5	4	subrngrng 20480	. . 3 ⊢ (𝐼 ∈ (SubRng‘𝑅) → 𝐽 ∈ Rng)
6	3, 5	ax-mp 5	. 2 ⊢ 𝐽 ∈ Rng
7		1z 12616	. . . . 5 ⊢ 1 ∈ ℤ
8		c0ex 11232	. . . . . 6 ⊢ 0 ∈ V
9	8	snid 4660	. . . . 5 ⊢ 0 ∈ {0}
10	7, 9	opelxpii 5710	. . . 4 ⊢ ⟨1, 0⟩ ∈ (ℤ × {0})
11	4	subrngbas 20484	. . . . . 6 ⊢ (𝐼 ∈ (SubRng‘𝑅) → 𝐼 = (Base‘𝐽))
12	3, 11	ax-mp 5	. . . . 5 ⊢ 𝐼 = (Base‘𝐽)
13	12, 2	eqtr3i 2758	. . . 4 ⊢ (Base‘𝐽) = (ℤ × {0})
14	10, 13	eleqtrri 2828	. . 3 ⊢ ⟨1, 0⟩ ∈ (Base‘𝐽)
15		oveq1 7421	. . . . . 6 ⊢ (𝑖 = ⟨1, 0⟩ → (𝑖(._r‘𝐽)𝑥) = (⟨1, 0⟩(._r‘𝐽)𝑥))
16	15	eqeq1d 2730	. . . . 5 ⊢ (𝑖 = ⟨1, 0⟩ → ((𝑖(._r‘𝐽)𝑥) = 𝑥 ↔ (⟨1, 0⟩(._r‘𝐽)𝑥) = 𝑥))
17	16	ovanraleqv 7438	. . . 4 ⊢ (𝑖 = ⟨1, 0⟩ → (∀𝑥 ∈ (Base‘𝐽)((𝑖(._r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(._r‘𝐽)𝑖) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐽)((⟨1, 0⟩(._r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(._r‘𝐽)⟨1, 0⟩) = 𝑥)))
18		id 22	. . . 4 ⊢ (⟨1, 0⟩ ∈ (Base‘𝐽) → ⟨1, 0⟩ ∈ (Base‘𝐽))
19	12	eleq2i 2821	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 ↔ 𝑥 ∈ (Base‘𝐽))
20	1, 2, 4	pzriprnglem6 21405	. . . . . . 7 ⊢ (𝑥 ∈ 𝐼 → ((⟨1, 0⟩(._r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(._r‘𝐽)⟨1, 0⟩) = 𝑥))
21	19, 20	sylbir 234	. . . . . 6 ⊢ (𝑥 ∈ (Base‘𝐽) → ((⟨1, 0⟩(._r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(._r‘𝐽)⟨1, 0⟩) = 𝑥))
22	21	a1i 11	. . . . 5 ⊢ (⟨1, 0⟩ ∈ (Base‘𝐽) → (𝑥 ∈ (Base‘𝐽) → ((⟨1, 0⟩(._r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(._r‘𝐽)⟨1, 0⟩) = 𝑥)))
23	22	ralrimiv 3141	. . . 4 ⊢ (⟨1, 0⟩ ∈ (Base‘𝐽) → ∀𝑥 ∈ (Base‘𝐽)((⟨1, 0⟩(._r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(._r‘𝐽)⟨1, 0⟩) = 𝑥))
24	17, 18, 23	rspcedvdw 3611	. . 3 ⊢ (⟨1, 0⟩ ∈ (Base‘𝐽) → ∃𝑖 ∈ (Base‘𝐽)∀𝑥 ∈ (Base‘𝐽)((𝑖(._r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(._r‘𝐽)𝑖) = 𝑥))
25	14, 24	ax-mp 5	. 2 ⊢ ∃𝑖 ∈ (Base‘𝐽)∀𝑥 ∈ (Base‘𝐽)((𝑖(._r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(._r‘𝐽)𝑖) = 𝑥)
26		eqid 2728	. . 3 ⊢ (Base‘𝐽) = (Base‘𝐽)
27		eqid 2728	. . 3 ⊢ (._r‘𝐽) = (._r‘𝐽)
28	26, 27	isringrng 20216	. 2 ⊢ (𝐽 ∈ Ring ↔ (𝐽 ∈ Rng ∧ ∃𝑖 ∈ (Base‘𝐽)∀𝑥 ∈ (Base‘𝐽)((𝑖(._r‘𝐽)𝑥) = 𝑥 ∧ (𝑥(._r‘𝐽)𝑖) = 𝑥)))
29	6, 25, 28	mpbir2an 710	1 ⊢ 𝐽 ∈ Ring

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 ∃wrex 3066 {csn 4624 ⟨cop 4630 × cxp 5670 ‘cfv 6542 (class class class)co 7414 0cc0 11132 1c1 11133 ℤcz 12582 Basecbs 17173 ↾_s cress 17202 ._rcmulr 17227 ×_s cxps 17481 Rngcrng 20085 Ringcrg 20166 SubRngcsubrng 20475 ℤ_ringczring 21365

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-addf 11211 ax-mulf 11212

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9459 df-inf 9460 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17416 df-prds 17422 df-imas 17483 df-xps 17485 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-subg 19071 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-cring 20169 df-subrng 20476 df-subrg 20501 df-cnfld 21273 df-zring 21366

This theorem is referenced by: pzriprnglem9 21408 pzriprngALT 21414 pzriprng1ALT 21415

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