Theorem rngorn1eq 38869

Description: Obsolete theorem. In a unital ring the range of the addition equals the range of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
rnplrnml0.1	⊢ 𝐻 = (2nd ‘𝑅)
rnplrnml0.2	⊢ 𝐺 = (1st ‘𝑅)

Assertion

Ref	Expression
rngorn1eq	⊢ (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)

Proof of Theorem rngorn1eq

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rnplrnml0.2	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2		rnplrnml0.1	. . . 4 ⊢ 𝐻 = (2nd ‘𝑅)
3		eqid 2760	. . . 4 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	rngosm 38835	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
5	1, 2, 3	rngoi 38834	. . . 4 ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) ∧ (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
6	5	simprrd 786	. . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
7		rngmgmbs4 38866	. . 3 ⊢ ((𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) → ran 𝐻 = ran 𝐺)
8	4, 6, 7	syl2anc 596	. 2 ⊢ (𝑅 ∈ RingOps → ran 𝐻 = ran 𝐺)
9	8	eqcomd 2766	1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   × cxp 5648  ran crn 5651  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  AbelOpcablo 31146  RingOpscrngo 38829

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-ov 7423  df-1st 8001  df-2nd 8002  df-rngo 38830

This theorem is used by:  rngoidmlem  38871  rngo1cl  38874  isdrngo2  38893