Theorem rngorn1eq 38438

Description: 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.)

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 2764	. . . 4 ⊢ ran 𝐺 = ran 𝐺
4	1, 2, 3	rngosm 38404	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
5	1, 2, 3	rngoi 38403	. . . 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 783	. . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
7		rngmgmbs4 38435	. . 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 593	. 2 ⊢ (𝑅 ∈ RingOps → ran 𝐻 = ran 𝐺)
9	8	eqcomd 2770	1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078  ∃wrex 3088   × cxp 5647  ran crn 5650  ⟶wf 6519  ‘cfv 6523  (class class class)co 7398  1^st c1st 7970  2^nd c2nd 7971  AbelOpcablo 30749  RingOpscrngo 38398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fo 6529  df-fv 6531  df-ov 7401  df-1st 7972  df-2nd 7973  df-rngo 38399

This theorem is referenced by:  rngoidmlem  38440  rngo1cl  38443  isdrngo2  38462