Theorem divmulsw 28609

Description: Relationship between surreal division and multiplication. Weak version that does not assume reciprocals. Later, when we prove precsex 28634, we can eliminate the existence hypothesis (see divmuls 28637). (Contributed by Scott Fenton, 12-Mar-2025.)

Assertion

Ref	Expression
divmulsw	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem divmulsw

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		divsval 28605	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → (𝐴 /su 𝐶) = (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴))
2	1	eqeq1d 2765	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐶 ∈ No ∧ 𝐶 ≠ 0s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
3	2	3expb 1138	. . . 4 ⊢ ((𝐴 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
4	3	3adant2 1149	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
5	4	adantr 486	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
6		simpl2 1211	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → 𝐵 ∈ No)
7		simp3l 1220	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ∈ No)
8		simp3r 1221	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐶 ≠ 0s )
9		simp1 1154	. . . . 5 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → 𝐴 ∈ No)
10	7, 8, 9	3jca 1146	. . . 4 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) → (𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No))
11		noreceuw 28607	. . . 4 ⊢ (((𝐶 ∈ No ∧ 𝐶 ≠ 0s ∧ 𝐴 ∈ No) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
12	10, 11	sylan 592	. . 3 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴)
13		oveq2 7451	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐶 ·s 𝑦) = (𝐶 ·s 𝐵))
14	13	eqeq1d 2765	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝐶 ·s 𝑦) = 𝐴 ↔ (𝐶 ·s 𝐵) = 𝐴))
15	14	riota2 7425	. . 3 ⊢ ((𝐵 ∈ No ∧ ∃!𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
16	6, 12, 15	syl2anc 596	. 2 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐶 ·s 𝐵) = 𝐴 ↔ (℩𝑦 ∈ No (𝐶 ·s 𝑦) = 𝐴) = 𝐵))
17	5, 16	bitr4d 285	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ (𝐶 ∈ No ∧ 𝐶 ≠ 0s )) ∧ ∃𝑥 ∈ No (𝐶 ·s 𝑥) = 1s ) → ((𝐴 /su 𝐶) = 𝐵 ↔ (𝐶 ·s 𝐵) = 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090  ∃!wreu 3367  ℩crio 7399  (class class class)co 7443  Nocsur 28027   0_s c0s 28221   1_s c1s 28222   ·_s cmuls 28522   /_su cdivs 28603

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-muls 28523  df-divs 28604

This theorem is used by:  divmulswd  28610  divs1  28620  divmuls  28637