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Mirrors > Home > MPE Home > Th. List > Mathboxes > 1t1e1ALT | Structured version Visualization version GIF version |
Description: Alternate proof of 1t1e1 11800 using a different set of axioms (add ax-mulrcl 10600, ax-i2m1 10605, ax-1ne0 10606, ax-rrecex 10609 and remove ax-resscn 10594, ax-mulcom 10601, ax-mulass 10603, ax-distr 10604). (Contributed by Steven Nguyen, 20-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
1t1e1ALT | ⊢ (1 · 1) = 1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1re 10641 | . 2 ⊢ 1 ∈ ℝ | |
2 | ax-1rid 10607 | . 2 ⊢ (1 ∈ ℝ → (1 · 1) = 1) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (1 · 1) = 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 (class class class)co 7156 ℝcr 10536 1c1 10538 · cmul 10542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-mulcl 10599 ax-mulrcl 10600 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rrecex 10609 ax-cnre 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 |
This theorem is referenced by: nnmul1com 39184 remulinvcom 39268 |
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