rngohom1 - Mathbox for Jeff Madsen

	Mathbox for Jeff Madsen		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > rngohom1		Structured version Visualization version GIF version

Theorem rngohom1 36126

Description: A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.)

**Hypotheses**
Ref	Expression
rnghom1.1	⊢ 𝐻 = (2^nd ‘𝑅)
rnghom1.2	⊢ 𝑈 = (GId‘𝐻)
rnghom1.3	⊢ 𝐾 = (2^nd ‘𝑆)
rnghom1.4	⊢ 𝑉 = (GId‘𝐾)

**Assertion**
Ref	Expression
rngohom1	⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉)

Proof of Theorem rngohom1 Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		rnghom1.1	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2738	. . . . 5 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4		rnghom1.2	. . . . 5 ⊢ 𝑈 = (GId‘𝐻)
5		eqid 2738	. . . . 5 ⊢ (1^st ‘𝑆) = (1^st ‘𝑆)
6		rnghom1.3	. . . . 5 ⊢ 𝐾 = (2^nd ‘𝑆)
7		eqid 2738	. . . . 5 ⊢ ran (1^st ‘𝑆) = ran (1^st ‘𝑆)
8		rnghom1.4	. . . . 5 ⊢ 𝑉 = (GId‘𝐾)
9	1, 2, 3, 4, 5, 6, 7, 8	isrngohom 36123	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆) ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦))))))
10	9	biimpa 477	. . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:ran (1^st ‘𝑅)⟶ran (1^st ‘𝑆) ∧ (𝐹‘𝑈) = 𝑉 ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝐹‘(𝑥(1^st ‘𝑅)𝑦)) = ((𝐹‘𝑥)(1^st ‘𝑆)(𝐹‘𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹‘𝑥)𝐾(𝐹‘𝑦)))))
11	10	simp2d 1142	. 2 ⊢ (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉)
12	11	3impa 1109	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹‘𝑈) = 𝑉)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ran crn 5590 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 GIdcgi 28852 RingOpscrngo 36052 RngHom crnghom 36118

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-map 8617 df-rngohom 36121

This theorem is referenced by: rngohomco 36132 rngoisocnv 36139

W3C validator