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Theorem rngisom1 20394
Description: If there is a non-unital ring isomorphism between a unital ring and a non-unital ring, then the function value of the ring unity of the unital ring is a ring unity of the non-unital ring. (Contributed by AV, 27-Feb-2025.)
Hypotheses
Ref Expression
rngisom1.1 1 = (1r𝑅)
rngisom1.b 𝐵 = (Base‘𝑆)
rngisom1.t · = (.r𝑆)
Assertion
Ref Expression
rngisom1 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∀𝑥𝐵 (((𝐹1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹1 )) = 𝑥))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑅   𝑥,𝑆
Allowed substitution hints:   𝐵(𝑥)   · (𝑥)   1 (𝑥)

Proof of Theorem rngisom1
StepHypRef Expression
1 rngimcnv 20384 . . . . . . . . 9 (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹 ∈ (𝑆 RngIso 𝑅))
2 rngisom1.b . . . . . . . . . 10 𝐵 = (Base‘𝑆)
3 eqid 2727 . . . . . . . . . 10 (Base‘𝑅) = (Base‘𝑅)
42, 3rngimrnghm 20383 . . . . . . . . 9 (𝐹 ∈ (𝑆 RngIso 𝑅) → 𝐹 ∈ (𝑆 RngHom 𝑅))
51, 4syl 17 . . . . . . . 8 (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹 ∈ (𝑆 RngHom 𝑅))
653ad2ant3 1133 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑆 RngHom 𝑅))
76adantr 480 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → 𝐹 ∈ (𝑆 RngHom 𝑅))
8 rngisom1.1 . . . . . . . . 9 1 = (1r𝑅)
98, 2rngisomfv1 20393 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹1 ) ∈ 𝐵)
1093adant2 1129 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹1 ) ∈ 𝐵)
1110adantr 480 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹1 ) ∈ 𝐵)
12 simpr 484 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → 𝑥𝐵)
13 rngisom1.t . . . . . . 7 · = (.r𝑆)
14 eqid 2727 . . . . . . 7 (.r𝑅) = (.r𝑅)
152, 13, 14rnghmmul 20377 . . . . . 6 ((𝐹 ∈ (𝑆 RngHom 𝑅) ∧ (𝐹1 ) ∈ 𝐵𝑥𝐵) → (𝐹‘((𝐹1 ) · 𝑥)) = ((𝐹‘(𝐹1 ))(.r𝑅)(𝐹𝑥)))
167, 11, 12, 15syl3anc 1369 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘((𝐹1 ) · 𝑥)) = ((𝐹‘(𝐹1 ))(.r𝑅)(𝐹𝑥)))
1716fveq2d 6895 . . . 4 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘(𝐹‘((𝐹1 ) · 𝑥))) = (𝐹‘((𝐹‘(𝐹1 ))(.r𝑅)(𝐹𝑥))))
183, 2rngimf1o 20382 . . . . . 6 (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:(Base‘𝑅)–1-1-onto𝐵)
19183ad2ant3 1133 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:(Base‘𝑅)–1-1-onto𝐵)
20 simpl2 1190 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → 𝑆 ∈ Rng)
212, 13rngcl 20095 . . . . . 6 ((𝑆 ∈ Rng ∧ (𝐹1 ) ∈ 𝐵𝑥𝐵) → ((𝐹1 ) · 𝑥) ∈ 𝐵)
2220, 11, 12, 21syl3anc 1369 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → ((𝐹1 ) · 𝑥) ∈ 𝐵)
23 f1ocnvfv2 7280 . . . . 5 ((𝐹:(Base‘𝑅)–1-1-onto𝐵 ∧ ((𝐹1 ) · 𝑥) ∈ 𝐵) → (𝐹‘(𝐹‘((𝐹1 ) · 𝑥))) = ((𝐹1 ) · 𝑥))
2419, 22, 23syl2an2r 684 . . . 4 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘(𝐹‘((𝐹1 ) · 𝑥))) = ((𝐹1 ) · 𝑥))
253, 8ringidcl 20191 . . . . . . . . . . . 12 (𝑅 ∈ Ring → 1 ∈ (Base‘𝑅))
26253ad2ant1 1131 . . . . . . . . . . 11 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 1 ∈ (Base‘𝑅))
2719, 26jca 511 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → (𝐹:(Base‘𝑅)–1-1-onto𝐵1 ∈ (Base‘𝑅)))
2827adantr 480 . . . . . . . . 9 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹:(Base‘𝑅)–1-1-onto𝐵1 ∈ (Base‘𝑅)))
29 f1ocnvfv1 7279 . . . . . . . . 9 ((𝐹:(Base‘𝑅)–1-1-onto𝐵1 ∈ (Base‘𝑅)) → (𝐹‘(𝐹1 )) = 1 )
3028, 29syl 17 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘(𝐹1 )) = 1 )
3130oveq1d 7429 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → ((𝐹‘(𝐹1 ))(.r𝑅)(𝐹𝑥)) = ( 1 (.r𝑅)(𝐹𝑥)))
32 simpl1 1189 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → 𝑅 ∈ Ring)
332, 3rngimf1o 20382 . . . . . . . . . . . 12 (𝐹 ∈ (𝑆 RngIso 𝑅) → 𝐹:𝐵1-1-onto→(Base‘𝑅))
34 f1of 6833 . . . . . . . . . . . 12 (𝐹:𝐵1-1-onto→(Base‘𝑅) → 𝐹:𝐵⟶(Base‘𝑅))
3533, 34syl 17 . . . . . . . . . . 11 (𝐹 ∈ (𝑆 RngIso 𝑅) → 𝐹:𝐵⟶(Base‘𝑅))
361, 35syl 17 . . . . . . . . . 10 (𝐹 ∈ (𝑅 RngIso 𝑆) → 𝐹:𝐵⟶(Base‘𝑅))
37363ad2ant3 1133 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:𝐵⟶(Base‘𝑅))
3837ffvelcdmda 7088 . . . . . . . 8 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹𝑥) ∈ (Base‘𝑅))
393, 14, 8, 32, 38ringlidmd 20197 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → ( 1 (.r𝑅)(𝐹𝑥)) = (𝐹𝑥))
4031, 39eqtrd 2767 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → ((𝐹‘(𝐹1 ))(.r𝑅)(𝐹𝑥)) = (𝐹𝑥))
4140fveq2d 6895 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘((𝐹‘(𝐹1 ))(.r𝑅)(𝐹𝑥))) = (𝐹‘(𝐹𝑥)))
42 f1ocnvfv2 7280 . . . . . 6 ((𝐹:(Base‘𝑅)–1-1-onto𝐵𝑥𝐵) → (𝐹‘(𝐹𝑥)) = 𝑥)
4319, 42sylan 579 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘(𝐹𝑥)) = 𝑥)
4441, 43eqtrd 2767 . . . 4 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘((𝐹‘(𝐹1 ))(.r𝑅)(𝐹𝑥))) = 𝑥)
4517, 24, 443eqtr3d 2775 . . 3 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → ((𝐹1 ) · 𝑥) = 𝑥)
4613ad2ant3 1133 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑆 RngIso 𝑅))
4746, 4syl 17 . . . . . . . 8 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑆 RngHom 𝑅))
4847adantr 480 . . . . . . 7 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → 𝐹 ∈ (𝑆 RngHom 𝑅))
492, 13, 14rnghmmul 20377 . . . . . . 7 ((𝐹 ∈ (𝑆 RngHom 𝑅) ∧ 𝑥𝐵 ∧ (𝐹1 ) ∈ 𝐵) → (𝐹‘(𝑥 · (𝐹1 ))) = ((𝐹𝑥)(.r𝑅)(𝐹‘(𝐹1 ))))
5048, 12, 11, 49syl3anc 1369 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘(𝑥 · (𝐹1 ))) = ((𝐹𝑥)(.r𝑅)(𝐹‘(𝐹1 ))))
5130oveq2d 7430 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → ((𝐹𝑥)(.r𝑅)(𝐹‘(𝐹1 ))) = ((𝐹𝑥)(.r𝑅) 1 ))
523, 14, 8, 32, 38ringridmd 20198 . . . . . 6 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → ((𝐹𝑥)(.r𝑅) 1 ) = (𝐹𝑥))
5350, 51, 523eqtrd 2771 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘(𝑥 · (𝐹1 ))) = (𝐹𝑥))
5453fveq2d 6895 . . . 4 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘(𝐹‘(𝑥 · (𝐹1 )))) = (𝐹‘(𝐹𝑥)))
552, 13rngcl 20095 . . . . . 6 ((𝑆 ∈ Rng ∧ 𝑥𝐵 ∧ (𝐹1 ) ∈ 𝐵) → (𝑥 · (𝐹1 )) ∈ 𝐵)
5620, 12, 11, 55syl3anc 1369 . . . . 5 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝑥 · (𝐹1 )) ∈ 𝐵)
57 f1ocnvfv2 7280 . . . . 5 ((𝐹:(Base‘𝑅)–1-1-onto𝐵 ∧ (𝑥 · (𝐹1 )) ∈ 𝐵) → (𝐹‘(𝐹‘(𝑥 · (𝐹1 )))) = (𝑥 · (𝐹1 )))
5819, 56, 57syl2an2r 684 . . . 4 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝐹‘(𝐹‘(𝑥 · (𝐹1 )))) = (𝑥 · (𝐹1 )))
5954, 58, 433eqtr3d 2775 . . 3 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (𝑥 · (𝐹1 )) = 𝑥)
6045, 59jca 511 . 2 (((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) ∧ 𝑥𝐵) → (((𝐹1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹1 )) = 𝑥))
6160ralrimiva 3141 1 ((𝑅 ∈ Ring ∧ 𝑆 ∈ Rng ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → ∀𝑥𝐵 (((𝐹1 ) · 𝑥) = 𝑥 ∧ (𝑥 · (𝐹1 )) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wral 3056  ccnv 5671  wf 6538  1-1-ontowf1o 6541  cfv 6542  (class class class)co 7414  Basecbs 17171  .rcmulr 17225  Rngcrng 20083  1rcur 20112  Ringcrg 20164   RngHom crnghm 20362   RngIso crngim 20363
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 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  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-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-er 8718  df-map 8838  df-en 8956  df-dom 8957  df-sdom 8958  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-sets 17124  df-slot 17142  df-ndx 17154  df-base 17172  df-plusg 17237  df-0g 17414  df-mgm 18591  df-mgmhm 18643  df-sgrp 18670  df-mnd 18686  df-grp 18884  df-ghm 19159  df-abl 19729  df-mgp 20066  df-rng 20084  df-ur 20113  df-ring 20166  df-rnghm 20364  df-rngim 20365
This theorem is referenced by:  rngisomring  20395  rngisomring1  20396
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