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Mirrors > Home > MPE Home > Th. List > zrhpropd | Structured version Visualization version GIF version |
Description: The ℤ ring homomorphism depends only on the ring attributes of a structure. (Contributed by Mario Carneiro, 15-Jun-2015.) |
Ref | Expression |
---|---|
zrhpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
zrhpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
zrhpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
zrhpropd.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
zrhpropd | ⊢ (𝜑 → (ℤRHom‘𝐾) = (ℤRHom‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2732 | . . . 4 ⊢ (𝜑 → (Base‘ℤring) = (Base‘ℤring)) | |
2 | zrhpropd.1 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
3 | zrhpropd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
4 | eqidd 2732 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘ℤring) ∧ 𝑦 ∈ (Base‘ℤring))) → (𝑥(+g‘ℤring)𝑦) = (𝑥(+g‘ℤring)𝑦)) | |
5 | zrhpropd.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
6 | eqidd 2732 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘ℤring) ∧ 𝑦 ∈ (Base‘ℤring))) → (𝑥(.r‘ℤring)𝑦) = (𝑥(.r‘ℤring)𝑦)) | |
7 | zrhpropd.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
8 | 1, 2, 1, 3, 4, 5, 6, 7 | rhmpropd 20330 | . . 3 ⊢ (𝜑 → (ℤring RingHom 𝐾) = (ℤring RingHom 𝐿)) |
9 | 8 | unieqd 4906 | . 2 ⊢ (𝜑 → ∪ (ℤring RingHom 𝐾) = ∪ (ℤring RingHom 𝐿)) |
10 | eqid 2731 | . . 3 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾) | |
11 | 10 | zrhval 20967 | . 2 ⊢ (ℤRHom‘𝐾) = ∪ (ℤring RingHom 𝐾) |
12 | eqid 2731 | . . 3 ⊢ (ℤRHom‘𝐿) = (ℤRHom‘𝐿) | |
13 | 12 | zrhval 20967 | . 2 ⊢ (ℤRHom‘𝐿) = ∪ (ℤring RingHom 𝐿) |
14 | 9, 11, 13 | 3eqtr4g 2796 | 1 ⊢ (𝜑 → (ℤRHom‘𝐾) = (ℤRHom‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∪ cuni 4892 ‘cfv 6523 (class class class)co 7384 Basecbs 17116 +gcplusg 17169 .rcmulr 17170 RingHom crh 20181 ℤringczring 20928 ℤRHomczrh 20959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2702 ax-rep 5269 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 ax-cnex 11138 ax-resscn 11139 ax-1cn 11140 ax-icn 11141 ax-addcl 11142 ax-addrcl 11143 ax-mulcl 11144 ax-mulrcl 11145 ax-mulcom 11146 ax-addass 11147 ax-mulass 11148 ax-distr 11149 ax-i2m1 11150 ax-1ne0 11151 ax-1rid 11152 ax-rnegex 11153 ax-rrecex 11154 ax-cnre 11155 ax-pre-lttri 11156 ax-pre-lttrn 11157 ax-pre-ltadd 11158 ax-pre-mulgt0 11159 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3371 df-reu 3372 df-rab 3426 df-v 3468 df-sbc 3765 df-csb 3881 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-pss 3954 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-iun 4983 df-br 5133 df-opab 5195 df-mpt 5216 df-tr 5250 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5615 df-we 5617 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6280 df-ord 6347 df-on 6348 df-lim 6349 df-suc 6350 df-iota 6475 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7340 df-ov 7387 df-oprab 7388 df-mpo 7389 df-om 7830 df-2nd 7949 df-frecs 8239 df-wrecs 8270 df-recs 8344 df-rdg 8383 df-er 8677 df-map 8796 df-en 8913 df-dom 8914 df-sdom 8915 df-pnf 11222 df-mnf 11223 df-xr 11224 df-ltxr 11225 df-le 11226 df-sub 11418 df-neg 11419 df-nn 12185 df-2 12247 df-sets 17069 df-slot 17087 df-ndx 17099 df-base 17117 df-plusg 17182 df-0g 17359 df-mgm 18533 df-sgrp 18582 df-mnd 18593 df-mhm 18637 df-grp 18787 df-ghm 19042 df-mgp 19933 df-ur 19950 df-ring 20002 df-rnghom 20184 df-zrh 20963 |
This theorem is referenced by: znzrh 21008 |
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